536 Documenta Praehistorica XLVII (2020) Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration Bernhard Weninger 1, Kevan Edinborough2 1 Institute of Prehistory, University Cologne, Köln, DE b.weninger@uni-koeln.de 2 Melbourne Dental School, Faculty of Medicine, Dentistry and Health Sciences, The University of Melbourne, Victoria, AU kevan.edinborough@unimelb.edu.au ABSTRACT – Following some 30 years of radiocarbon research during which the mathematical prin- ciples of 14C-calibration have been on loan to Bayesian statistics, here they are returned to quantum physics. The return is based on recognition that 14C-calibration can be described as a Fourier trans- form. Following its introduction as such, there is need to reconceptualize the probabilistic 14C-analy- sis. The main change will be to replace the traditional (one-dimensional) concept of 14C-dating pro- bability by a two-dimensional probability. This is entirely analogous to the definition of probability in quantum physics, where the squared amplitude of a wave function defined in Hilbert space pro- vides a measurable probability of finding the corresponding particle at a certain point in time/space, the so-called Born rule. When adapted to the characteristics of 14C-calibration, as it turns out, the Fourier transform immediately accounts for practically all known so-called quantization properties of archaeological 14C-ages, such as clustering, age-shifting, and amplitude-distortion. This also applies to the frequently observed chronological lock-in properties of larger data sets, when analysed by Gaus- sian wiggle matching (on the 14C-scale) just as by Bayesian sequencing (on the calendar time-scale). Such domain-switching effects are typical for a Fourier transform. They can now be understood, and taken into account, by the application of concepts and interpretations that are central to quantum physics (e.g. wave diffraction, wave-particle duality, Heisenberg uncertainty, and the correspondence principle). What may sound complicated, at first glance, simplifies the construction of 14C-based chro- nologies. The new Fourier-based 14C-analysis supports chronological studies on previously unachiev- able geographic (continental) and temporal (Glacial-Holocene) scales; for example, by temporal se- quencing of hundreds of archaeological sites, simultaneously, with minimal need for development of archaeological prior hypotheses, other than those based on the geo-archaeological law of stratigra- phic superposition. As demonstrated in a variety of archaeological case studies, just one number, defined as a gauge-probability on a scale 0–100%, can be used to replace a stacked set of subjective Bayesian priors. KEY WORDS – radiocarbon calibration; Fourier transform; Born probability; Santorini DOI> 10.4312\dp.47.31 IZVLE∞EK – Po pribli∫no 30 letih radiokarbonskih raziskav, v katerih si je Bayesova statistika izpo- sojala matemati≠ne principe 14C-kalibracije, le-te sedaj vra≠amo v kvantno fiziko. Ta vrnitev je osno- vana na predpostavki, da lahko 14C-kalibriranje opi∏emo kot Fourierjevo transformacijo. In to ima za posledico, da je potrebno ponovno konceptualizirati verjetnostno analizo 14C. Poglavitna spre- memba bo zamenjava tradicionalnega (enodimenzionalnega) koncepta verjetnosti 14C-datiranja z dvodimenzionalno verjetnostjo. To je povsem analogono definiciji verjetnosti v kvantni fiziki, kjer kvadratna amplituda valovne funkcije, ki je definirana v Hilbertovem prostoru, zagotavlja merljivo Bayesova racionalnost 14C, Heisenbergovo na;elo nedolo;enosti in Fourierjeva transformacija> lepota radiokarbonske kalibracije Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 537 tioned above, the particular advantage of Bayesian modelling is that a wide spectrum of archaeological prior information can be included in the analysis. There is no question that Bayesian age-modelling can be recommended for virtually all fields of archa- eological research. On the other hand, there have been persistent rumblings of discontent in parallel to the general acceptance of this statistics driven ar- chaeological paradigm. One frequent expectation by archaeologists is that Bayesian modelling is faultless. It is seen to be capable of providing what appears to be universally coherent results, both in archaeolo- gical and mathematical terms, for all 14C-based stu- dies. Unfortunately, this expectation is in clear con- tradiction with the cautiousness, vigilance, and con- siderable experience needed for good Bayesian mo- delling, as described by Caitlin E. Buck and Bo Me- son (2015). Similar concerns with regard to the wide- spread and naive belief in Bayesian-based chronolo- gies are expressed by many of the authors of the World Archaeology Special Issue, Volume 47 (2015), that is dedicated to Bayesian radiocarbon chronolo- gy. The editorial conclusion of Paul Pettitt and João Zilhão (2015.526) is summarized as follows: “many existing models are faulty”. In the present paper, some five years later, we have reason to continue this discussion. Again, our aim is to optimize the application of radiocarbon dating in archaeological research. This time, however, it is not any particular archaeological case study, nor any Introduction The analysis of radiocarbon data, aimed at construc- tion of cultural chronologies at very high precision (even with the loss of accuracy), has been a conti- nuous field of research for many decades. Much ef- fort has been invested in the development of statis- tical models that allow incorporation of both quali- tative and quantitative archaeological information in the construction of 14C-chronologies. The large majority of these models utilize Bayesian theory (e.g., Buck et al. 1991; 1992; Nicholls, Jones 2000; Weninger F. et al. 2000; Bayliss 2009; Blaauw, Chri- sten 2011; Bronk Ramsey 2009; Steier, Rom 2000; Weninger F. 2011) and applications of Bayesian age-modelling still have growing networks in archa- eology, dendrochronology, terrestrial geomorpholo- gy, in ice-core studies and many other fields. In part this is due to the convenient availability of advanced software, such as OxCal (Bronk Ramsey 2020) BCal (Buck et al. 2020), CALIB (Stuiver et al. 2020), and Bacon (Blaauw, Christen 2011). But perhaps the main reason why Bayesian 14C-analysis today repre- sents the dominant paradigm (Buck, Meson 2015. 567; Buck, Juarez 2017.5) is that some 20 years ago the necessary statistical procedures were translated into computer language, so researchers today have at their disposal a well-established and now rigor- ously formalized statistical framework for chronolo- gical modelling (e.g., Bronk Ramsey 2009; 2020; Buck, Meson 2015). When applied to the construc- tion of 14C-based archaeological chronologies, aimed at the highest achievable dating precision, as men- verjetnost iskanja ustreznega delca v dolo≠eni to≠ki v ≠asu/prostoru, to je t.i. Bornovo pravilo. Fou- rierjeva transformacija, ki jo prilagodimo zna≠ilnostim 14C-kalibracije, se nemudoma prilagodi tako reko≠ vsem t.i. lastnostim kvantizacije arheolo∏kih 14C datumov, kot so hierarhi≠ne metode zdru∫e- vanja, spreminjanje starosti in popa≠enje amplitude. To velja tudi za pogosto opazovane kronolo∏ke lastnosti zaklepanja pri ve≠jih podatkovnih bazah, ≠e jih analiziramo z Gaussovim usklajevanjem krivulje (na lestvici 14C) tako kot z Bayesovim zaporedjem (v koledarskem ≠asovnem merilu). Tak- ∏ni u≠inki prekopa domene so zna≠ilni za Fourierjevo transformacijo. Zdaj jih lahko razumemo in upo∏tevamo z vpeljavo konceptov in interpretacij, ki so osrednjega pomena v kvantni fiziki (npr. di- frakcija valovnih dol∫in, dvojnost valov in delcev, Heisengergovo na≠elo nedolo≠enosti in princip ko- respondence). Kar se na prvi pogled zdi zapleteno, v resnici poenostavi postavitev 14C-kronologij. Nova 14C analiza, ki temelji na Fourierjevi transformaciji, podpira kronolo∏ke ∏tudije na prej nedo- segljivih geografskih (kontinentalnih) in ≠asovnih (v glacialih v holocenu) lestvicah; npr. s ≠asovno sekvenco na stotine arheolo∏kih najdi∏≠ hkrati, z minimalno potrebo po razvoju predhodnih arheo- lo∏kih hipotez, razen tistih, ki temeljijo na geo-arheolo∏kih zakonih stratigrafske superpozicije. Kot je razvidno iz razli≠nih arheolo∏kih ∏tudijskih primerov lahko tudi le z eno ∏tevilko, ki je definirana kot merilna verjetnost na lestvici od 0 do 100%, nadomestimo zlo∫en nabor subjektivnih Bayesovih apriornih verjetnosti. KLJU∞NE BESEDE – radiokarbonska kalibracija; Fourierjeva transformacija; Bornova verjetnost; Santorini Bernhard Weninger, Kevan Edinborough 538 assumption of maths-weakness within the archaeolo- gical user community, that drives the discourse, in- stead it is the very mathematical foundation of 14C- calibration. As an alternative to its suggested univer- sal implementation as a Bayesian paradigm (Hea- ton et al. 2020.4), in our view the process of 14C-ca- libration is best described as being a Fourier trans- form which thus has its mathematical foundation in Hilbert space theory. The main purpose of the pre- sent paper is to review the concepts of the Fourier transform and associated aspects of quantum theory, and describe it in an understandable manner, using an archaeological context to explain why these con- cepts can together provide a unique mathematical background to 14C-age calibration. Naturally, this program requires mathematical vali- dation. Simultaneously we wish to avoid the large- scale ceremonial presentation of corresponding pro- ofs, definitions, equations, formulas and theorems, although we derive a local minimum for all this in the Appendix. The first reason for this decision is that details of the Fourier transform are easily found in textbooks of quantum physics and optical or elec- tronic signal processing. For these topics there are many online presentations, wherein the required mathematics is didactically well-developed. A fur- ther reason is that the translation of 14C-calibration into the language of a Fourier transform is relatively straightforward, once a few points in the terminolo- gy have been clarified. Once this has been done, the very existence of so many telling analogies between 14C-calibration and quantum physics is itself suffi- cient to prove the point. Given that under the Fou- rier transform the calibration algorithms of CalPal- software remain unchanged – one of us (BW) has always been using their mathematical background in quantum theory in his work – any attempt to re- write the existing mathematics of the Fourier trans- form would be superfluous. On the other hand, given that our persistent observations of analogies be- tween 14C-ages and quantum particles have never received much attention in the radiocarbon commu- nity, perhaps due to deficits in mathematical forma- lization, or perhaps because it may sound like an outrageous idea, we now provide the reader with an appropriate selection of equations and so-called engineering rules (e.g., the Fourier transform of a Gaussian is again a Gaussian). Perhaps thankfully, the archaeological reader does not have to become deeply acquainted with the underlying mathematics, nor of Heisenberg’s uncertainty principle. Instead, such a reader only needs to grasp the generalized re- sult here, which is that practically all observed quan- tum properties of archaeological 14C-ages can now be understood as what they are, the mathematical consequence of a Fourier transform. A simple notion of mathematical foundation To be as clear as possible, 14C-calibration is so much a Fourier transform that we have taken the freedom, in one or the other figure captions below, to replace the very expression 14C-calibration with Fourier transform. It goes without saying that this equiva- lence in mathematical background applies equally to what is presently known as Bayesian 14C-calibration, sequencing, age-depth modelling and so on, just as for non-Bayesian wiggle matching, barcode seria- tion, or construction of summed calibrated probabi- lity distributions. Despite the existence of these many different methods, variants, and names, and whether the calculations are technically designed to run on one (or the other) of the two scales, or switch be- tween the two domains, or whether additional data, or hypotheses, are included in the analysis, all these methods are fundamentally identical in the sense that, ultimately, they are all based on a Fourier trans- form. The solution we adapt in describing this re- sult is to a large part historical analysis in the main text, in combination with a technical description in the Appendix. Some carefully selected rules of trans- lation, as provided in the Appendix, are themselves designed to mimic a Fourier transform: they trans- late backwards and forwards between radiocarbon dating in traditional terminology, and 14C-quantum language in the new perspective. Our hope is that these translation rules may be helpful beyond the present study, in case the reader wishes to validate and extend these new concepts. The report structure The overall structure of our report is as follows. To begin, we take freedom to deconstruct the rather naive notion, upheld by many archaeologists, that (likelihood-ratio based) Bayesian 14C-theory can be understood as the unconditional ultima ratio imple- mentation of radiocarbon dating in archaeology. We do this by reference to the history of gravitational theory, which has shown wonderful advances over many centuries, but without need, necessity, or even knowledge of Bayes’ theorem. This suggests that, since Bayes’ probability theory has not been parti- cularly fundamental for this kind of important sci- entific research in the past, this might also be the case for 14C-analysis in the future. Following this prelude, we proceed towards the somewhat less tri- Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 539 vial deconstruction of the – supposed – mathemati- cal foundation of 14C-calibration in Bayes’ theorem, put forward some 27 years ago by Herold Dehling and Johannes van der Plicht (1993). The two stud- ies, in combination, are aimed at providing a slow and protracted introduction of the new mathemati- cal concepts, hence demonstrating at least the possi- bility that there might exist a world beyond Bayes. The point hereby is that, following some 30 years of dedicated, persistent, persuasive and sometimes even convincing Bayesian didactics, parts of the ar- chaeological community have developed academic traits that future prehistorians could possibly de- scribe as Bayesian entanglement. Having motivated both the necessity and possibility for a change in the mathematical foundation of the 14C-calibration procedure, there comes the point where we must actually depart from the world of Bayesian 14C-ana- lysis. Unfortunately, there does not appear to exist a slow and continuous transition from Bayes-based proba- bility theory to the quantum theory needed in 14C- analysis. Notwithstanding all that has been achieved in the last 30 years, Bayesian 14C-analysis now ap- pears to us to be an increasingly reckless journey, with routine passage yet moving at rather too high speed along a not well-constructed road. In compa- rison, and judging from the experience as formulat- ed in practically all text-books, the transition from classical physics to quantum theory is inevitably ab- rupt. Quite simply, there is no slow and easy transi- tion. One accustoms oneself to the new concepts, which takes a bit of time, and soon forgets how much the world has changed. To motivate the read- er towards the realization that this transition is be- neficial, and also in the context of 14C-analysis, we provide a review of the paper by John Skilling and Kevin H. Knuth, entitled ‘The Symmetrical Founda- tion of Measure, Probability, and Quantum Theory’ (Skilling, Knuth 2019). New concepts of probabilistic 14C-analysis Then, having introduced (in the Appendix) the 14C- calibration as a Fourier transform, which is imme- diately operative in an abstract vector (so-called: Hil- bert) space, we are prepared for the corresponding conceptual switch in probabilistic 14C-analysis. The main change will be to replace the traditional (one- dimensional) concept of 14C-dating probability by a two-dimensional probability. This is entirely analo- gous to the corresponding definition of probability as used in quantum physics, where the (measurable) probability of finding a wave/particle at a certain point in time/space is based on the squared magni- tude of its amplitude: the so-called Born rule. When adapted to the needs of 14C-analysis, application of the modulus-squared Born rule leads us to recognize the 14C-dating probability is not well described as an area under the curve of the graph that shows the summed calibrated Gaussians. Instead, and in mathe- matically more satisfactory terms, it is possible (for the special purposes of archaeological 14C-analysis) to define a Fourier-based dating probability as pro- duct of the amplitudes of two distributions given on the two scales (i.e. 14C and calendric). Formulas for the 14C-based Born probability are given in the Ap- pendix [10]. In principle, what has up to now been termed a calibrated probability distribution (CPD) we now call a wave-function. This terminology is chosen due to the incomplete scaling properties of the calibrated distribution. The actually measurable (what we call gauged) dating probability is instead represented as a rectangular (two-dimensional) area that can be projected onto the calibration curve. But we can project such probability rectangles, if re- quested, onto other components of the calibration system. The flexibility of the new gauge-probability concept is illustrated first in Figure 1, by construc- tion of the entire sequence of major calibration curve plateaus for INTCAL20, then again in Figure 2, by gauge-seriation of the recently published strati- fied Gravettian 14C-data from Abri Pataud (Douka et al. 2019), but without the need for stratification mo- delling, and finally in Figure 3, by showing the wave- function (alias CPD) for 2543 14C-ages from Greece and Crete (data: Katsianis et al. 2020) in graphic comparison with the underlying (N=303) archaeolo- gical sites, with automated site-seriation. As for the first example (Fig. 1), up to now the pla- teaus of the calibration curve have been known mainly for their precision-limiting properties, which were qualitatively defined. When quantified, as it turns out, the plateaus have all the necessary pro- perties to be interpreted as measurable (modulus squared) dating probabilities in Born’s sense. As for the second example (Fig. 2), we note that in a typi- cal archaeological 14C-data set there are usually many open (i.e. not automatically given) probability measures, but which can be chosen (i.e. gauged) by the observer. As illustrated here for the Gravettian sequence at Abri Pataud, already by the simple measure of cho- osing one gauge for the total data we achieve a four- fold separation of the underlying archaeological Bernhard Weninger, Kevan Edinborough 540 units. The methodological point of interest here is that the choice of a gauge meets the expectation of a probability definition, but without need for its in- terpretation as an archaeological hypothesis. The gauge is no more, nor less, than an explorative tool. What we have done, in this case, is to slowly (on- screen) move the gauge-level up and down an arbi- trary gauge scale (0 £ g £ 100%), in search of a gauge-value for which the corresponding rectangles just start touching each other. The results are in im- mediate agreement with the thoroughly detailed chrono-stratigraphic analysis by Katerina Douka et al. (2020). Importantly, qua construction method, for whatever gauge we choose, the rectangles cannot overlap, nor can they occupy the same area. In mea- sure theory, what we call ‘gauging’ is a standard pro- cedure used to assign Lebesgue measures (in place of probabilities) to subsets of the study data. As a third illustration of the new Fourier-based dating concepts, Figure 3 shows the 14C-demographic chro- nology of Greece (including Crete and the Aegean islands) for the last 12 000 years, based on the re- cently published database of Antonio Katsianis et al. (2020). Of particular interest, we note (1), the broad- ly synchronous end of the eastern Mediterranean wet period of Sapropel S1 and abrupt onset of Ra- pid Climate Change (RCC) conditions (~6.2 ka cal BP), with the major settlement gap (6.2–5.0 ka cal BP), that is very clearly observable not only in the summed 14C-data, but also on site level (cf. Wenin- ger et al. 2009), as well as (2), the need for further studies on methods of automated multi-phase site discrimination. To conclude this overview, all we are doing by intro- ducing these new physics-based concepts into the world of 14C-analysis is to fully account for the ma- thematical description of 14C-calibration as a Fourier transform. Within this mathematical setting there may be a need for technical changes in the algori- Fig. 1. Automated plateau-box construction for calibration curve INTCAL20 (green line) based on gauge integration in the age range 0–55 ka cal BP. Input: Dirac comb with N=55000 equidistant INTCAL20-de- rived samples measured at s=±100 BP. Output: 14C-histogram and back-calibrated summed probability distribution (SPD: black curve) with gauge-probabilities (red-lines) set at p=50% SPD-amplitude. Lower in- sert: NGRIP stable oxygen d18O isotope data on GICCO5 timescale (Anderson et al. 2006). Note the inte- resting correlation of certain plateau rectangles with onset of major Greenland Interstadials (cf. Wenin- ger 2020). Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 541 thms used in calibration software, other than in Cal- Pal, which is a matter for each reader to judge for themselves. As for changing the underpinning theo- ry, we can readily imagine a combined Fourier-Baye- sian statistical approach, whereby the calibration is performed on the basis of the Fourier transform, and then archaeological age-modelling is run within a Bayesian framework. In an engineering context, such a combined approach is viable, if it is based on tried and proven research where the priors are well understood and therefore less subjective. In prac- tice, the choice of specific likelihood functions need- ed to make Bayesian revisions to a given engineer- ing risk assessment remains difficult to justify. This is especially so when the supporting information for a given prior does not originate from a relatively straightforward process of statistical sampling (Wink- ler 1996). For example, although NASA sometimes uses specific likelihood functions as part of their high-profile assessments to minimize potentially ca- tastrophic space-flight related risks, the use of Bayes- based risk assessment usually occurs when the rele- vant engineering constraints involved are already clearly established, for instance after the thorough testing of key mechanical components. Somewhat surprisingly, however, our reading of the available literature suggests that probabilistic risk assessments made by NASA still do not rely heavily on Bayes- based statistical methods, despite huge increases in computational power now available since the Apol- lo program (Lutomski 2013). Bayesian 14C-rationality As mentioned in the introduction, some five years have passed since Pettitt and Zilhão (2015) conclud- ed that many existing Bayesian age models are faulty. Our present topic, however, is something very different. Do not be concerned, we will not be scan- ning the increasingly vast Bayesian 14C-literature in search of some slightly imperfect age-model, let alone in pursuit of some unfortunate sub-optimal ar- chaeological prior. Instead, we will be addressing the one important question that has not yet been widely studied: why must the application of radiocarbon dating always (under all conditions) be based on Ba- yesian research methodology? Isaac Newton’s hypotheses hon fingo As stated above, the Fourier transform is a promis- ing alternative to Bayes, not only in technical but also conceptual terms. An example, recently demon- strated in Bernhard Weninger (2020), is the possibi- lity to perform automated seriation of hundreds of 14C-dated archaeological sites (assuming the under- lying data is truly representative of said sites), with- out need for development of further archaeological hypotheses post-excavation. In terms of in many cases expendable archaeological modelling, otherwise the very hallmark of Bayesian 14C-analysis, this is remi- niscent of the statement hypotheses non fingo by Isaac Newton, who wrote in an essay ‘General Scho- lium’ that was added to the second edition (1713) of his Mathematical Principles of Natural Philosophy (first edition: 1686 – commonly abbreviated Principia), which included the follo- wing: “Hitherto I have not been able to dis- cover the cause of those properties of gravity from phenomena, and I frame no hypotheses, hypotheses non Fig. 2. Application of the probabi- lity-gauge sequencing method to a Gravettian 14C-data set from Abri Pataud (Douka et al. 2020). Data in- put: N=27 14C-ages from Doukas et al. (2020.Fig. 4). Red rectangles: 2D- Born-probabilities for applied g=9% gauge. Blue rectangles: Results of Ba- yesian sequencing based on sample- grouping by different levels and stra- tigraphical level-modelling. Bernhard Weninger, Kevan Edinborough 542 Fig. 3. 14C-Demographic chronology of Greece and Crete based on data of Katsianis et al. (2020), in con- text with climate records, A Lake Neor (NW-Iran) dust flux as proxy for the Siberian High (Sharafi et al. 2015; cf. Mayewski et al. 1997) and for Rapid Climate Change (RCC) (most recently: Rohling et al. 2019); B GISP2 ice-core δ18O as proxy for Greenland surface air temperature Grootes et al. (1993); C N=2543 summed calibrated 14C-ages from Greece & Crete (database: Katsianis et al. 2020); D Barcode Seriation of 303 sites from same database (unfiltered; all sites with ≥3 dates shown) with gauge=40% and leading edge timing (cf. Weninger 2020). Vertical shading: periods with extreme climate variability. Upper hori- zontal shading: E-Mediterranean Sapropel S1 with subdivisions S1a & S1b as proxy for moist conditions, with dry/cold RCC-interruption 8.6–8.0 ka cal BP (cf. Schmiedl et al. 2010). Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 543 fingo; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechani- cal, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena and afterwards rendered ge- neral by induction.” Interestingly, this quite unusual description of what makes science rational (and clearly based on New- ton’s long experience with the topic), was formulat- ed by Newton some 100 years prior to Thomas Ba- yes’ publication of ‘An Essay Towards Solving a Problem in the Doctrine of Chances’ (1763). Ac- cording to the historical analysis by Abigail E. Bell (1942), Newton’s main intention with this statement is “to keep science clear of metaphysical entan- glements”. A more recent analysis of the hypotheses non fingo statement is by Scott Milner (2018), who makes the important point that certain disciplines may have good reason to not be hypothesis driven. Correspondingly, his paper is entitled ‘Newton Didn’t Frame Hypotheses – why should we?’ As an exam- ple, the rationality of which presumably only few would find reason to doubt, Scott Milner reminds us of Albert Einstein’s studies on gravitation. These were indeed (to begin with) so purely theoretical that – we may add – Einstein himself was apparent- ly quite astonished that his theory could actually be used to forecast a measurable astronomical effect, the advance of Mercury’s perihelion that was ob- served in 1915, where Einstein had not planned for his gravitation theory to measure such effects. Let us go a step further. The two approaches of Newton and Einstein are almost diametrically opposed. Ap- plying modern terminology, whereas Newton’s gra- vitational studies are largely experimental, Einstein’s studies are primarily theoretical (unless we decide that Gedankenexperiments are experiments). Hence, we have in both comparisons the unique chance to study the true potential of Bayes’ theorem, in terms of what would happen (Aristoteles’: teloV), should we swap the two sets of paired input variables, na- mely q=Theory and D=Data in Bayes’ theorem (cf Eq. 1). The result (see below for further discussion of the structure of Bayes’ theorem) is – nothing. Namely, for neither of the two aforementioned sci- entists does the application of Bayes’ theorem ap- pear to have been important. For Newton this is tri- vially true, by virtue of his having lived before Ba- yes. For Einstein we conclude the same, since for his research the application of Bayes’ theorem was either unnecessary, or using it never occurred to Einstein, which amounts to the same. The following quote is what Einstein actually writes in a paper sub- mitted to the Prussian Academy of Sciences in 1915 (i.e. prior to the famous experimental confirmation, in 1919, of Einstein’s predicted bending of light in strong gravitational fields): “In der vorliegenden Arbeit finde ich eine wich- tige Bestätigung dieser radikalsten Relativitätsthe- orie; es zeigt sich nämlich, daß sie die von LEVER- RIER entdeckte säkulare Drehung der Merkurbahn im Sinne der Bahnbewegung, welche etwa 45“ im Jahrhundert beträgt qualitativ und quantitativ er- klärt, ohne daß irgendwelche besondere Hypothe- se zugrunde gelegt werden müßte.“ (Einstein 1915; taken from von Meyenn 1990.234). “In this paper I find an important confirmation of this most radical relativity theory; it is shown, na- mely, that this theory explains both qualitatively and quantitatively the secular advance of Mer- cury’s orbital movement discovered by LEVERRIER, and which amounts to around 45” per century, without need for the formulation of any particular hypothesis.” (Our translation). Although Einstein’s version of the hypotheses non fingo statement is clearly different from Newton’s, it is similarly non-Bayesian. In physics, just as in bio- logy and other disciplines, it is not always helpful to first devise hypotheses and subsequently test them (unless we decide that Gedankenexperiments are hypotheses). This applies similarly to quantum string theory (as is well known), as well as to prehi- storic archaeology (as is less apparent), although one could argue that Bayes-based approaches to radio- carbon dating in archaeology are often more about parameter estimation than hypothesis testing, an in- teresting point that could be made much clearer in much more published research. Bayes’ theorem According to Bayes’ theorem (Eq. 1), in a formal re- presentation taken from Skilling and Knuth (2019), the probability (i.e. scaled truth values 0–100%) of achieved (output) posterior results is not only de- pendent on the truth value of the empirical (input) evidence, but also on the validity of the (input) prior belief. Both can be seen in normal scientific proce- dures, because yes, we learn by experience. What is more remarkable is the formal structure of Bayes’ theorem, and in particular its symmetry: P(q) · P(D|q) = P(D) · P(q|D) [Eq. 1] Prior·Likelihood = Evidence·Posterior Bayes’ theorem Bernhard Weninger, Kevan Edinborough 544 As it appears, the Bayes formula is entirely sym- metric in terms of probabilities p(q) and p(D) that are formulated for variables q=Theory and D=Data, as well as for the conditional probabilities p(q|D) and p(D|q). As a consequence, it would be easily possible – with no methodological restriction – to reformulate the Bayes formula, with a commuta- tive switch in the positions of variables q and D. We would then not only have q=Data and D=Theory, but the earlier priors would turn into evidence, the previous posteriors would become past likelihoods, and we could even swap their pairwise multiplica- tion, with no change at all in what is termed ‘Baye- sian inference’. This wonderfully open structure of Bayes’ theorem becomes yet more apparent when its four different components are introduced as sym- bolic variables. Let us call them A, B, C, D. It then follows that A·B = C·D. By solving Equation 1 for each of its variables, we recognize the existence of a total of four distinct possibilities of division by zero (A=(C·D)/B, or B=(C·D)/A, or C=D/(A·B), or D= A·B)/C), that we should avoid under all circumstan- ces. Beyond this seemingly minor caveat, the symmetric (commutative) Bayesian entanglement of ‘Theory’ and ‘Experiment’ is most remarkable and, to our knowledge, would find its closest analogue not in Popperian falsifiability (a widely assumed funda- ment of scientific rationality), but rather more in the more flexible (historiographic) scientific philosophies of Thomas Kuhn and Paul Feierabend. There are other philosophies of rationality that have accom- panied the development – in our case – of quantum theory, ranging from the Copenhagen interpreta- tion through Many worlds to Bohm’s Hidden-vari- able theory, and others. When applying Bayes’ theorem to 14C-applications in a single calibration the 14C-measurement (on the 14C-scale) is the likelihood function, and an updated probability is seen on the calendrical scale. In more complex Bayes-based 14C models the calendar date- ranges become the likelihood functions in the para- meter estimation process. Although this provides an extremely flexible methodological approach for many researchers, we urge that much more thought about this complex process is now required. The fun- damental scientific rationality that underlines the expression of Bayes’ theorem in radiocarbon calibra- tion is best illustrated by the following story. Once, when asked whether he truly believed that the hor- seshoe hung above his door would bring him luck, Nils Bohr apparently replied: “No, but I am told that they bring luck even to those who do not believe in them.” Lurking in the shadow of Bayes-based 14C- calibration, is the danger of wrongly applying mul- tiple normalization to distributions that only look as if they were defined on two independent time- scales. In reality, there is only one distribution (or horseshoe) which is defined on two domains. Symmetrical foundation of measure, proba- bility, and quantum theories Having updated Bayes’ theorem in perhaps a some- what idiosyncratic manner, but nevertheless in ad- miration of Bayesian rationality, we can make some further positive reference (despite remaining ca- veats) to the study of John Skilling and Kevin H. Knuth (2019), entitled Symmetrical foundation of measure, probability, and quantum theories. One of their claims – and this contrasts nicely with some- times more sophisticated ones – is that “there is no mystery or weirdness about quantum theory”. The supposedly simple reason for this finding is that quantum theory contains nothing but ordinary pro- babilistic inference. Well, although we would really like to accept such a mysterious statement, unfor- tunately Skilling and Knuth (2019), leave aside the discussion of the one single and altogether most im- portant property that all quantum theories have, and that is the noncommutativity of certain paired vari- ables, such as impulse/momentum and energy/time. We will return to the corresponding question of the 14C-related noncommutativity below. As for Bayes’ theorem, having stated its fundamental importance as being the foundation of rational inference, Skil- ling and Knuth (2019) conclude that it contains the “same simple laws of proportion that apply wide- ly elsewhere”. Indeed, that is exactly the structure A·B=C·D of the Bayes formula that we have recog- nized, above. Henceforth, and now accepting that both quantum theory and Bayes’ theory have cer- tain limits and restrictions (although in our view, Bayes may well have more of both), why not com- bine both methods? Such was the very proposal we put forward some time ago (Weninger et al. 2011). In the meantime, we must concede allowance for a change in opinion. Our recently derived notion is that a satisfactory mathematical foundation of 14C-calibration can in- deed be found in quantum theory, which no longer requires the expedient and computationally more expensive use of Bayes’ theorem. Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 545 What is non-classical probability theory? The earliest (detailed) suggestion that probabilistic 14C-calibration can be based upon Bayes’ theorem can be found in a seminal paper by Dehling and van der Plicht (1993), where it is based on the following arguments (ebda.244): ”Mathematical pitfalls can cause calibration pro- cedures to contradict classical formulas ... We show that these ambiguities can be understood in terms of classical and Bayesian approaches to statistical theory. The classical formulas correspond to a uni- form prior distribution along the BP axis, the [Ba- yesian] calibration procedure to a uniform prior distribution along the calendar axis. We argue that the latter is the correct choice, i.e. the [Bayesian] computer programs used for radiocarbon calibra- tion are correct.” We have here the proposal, that – thanks to Bayes – we may now relax and remain forever (as it were) reassured that the mathematical procedure of 14C- calibration is Bayesian, and is furthermore best ap- plied from the perspective of the calendar axis. From a technical perspective, we agree. This is an optimal and expedient approach. It avoids the problems of division by zero that inevitably occur when the cali- bration algorithm is run from a 14C-scale perspec- tive. To this point, however, we note that never once during some 35 years’ experience with the inverse calibration from the perspective of the calendric time-scale have we experienced a problem with di- vision by zero. This is despite the fact that, in 14C- calibration, there are all kinds of potentially threat- ening problems of this type (Bohr’s Horseshoe). The choice of an expedient algorithm, alone, does not make the underlying process Bayesian. Although it is described and even recommended as Bayesian by Dehling and van der Plicht (1993), in actual fact the algorithm is simply identical to the procedure described seven years earlier by Weninger (1986). In that paper, however, no mention is made of Ba- yes’ theorem. Instead, it mentions the existence of an unresolved normalization problem, one which still exists today, but which is now so deeply con- cealed inside Bayes’ theorem that is hiding in plain sight, producing one publication after another of cu- riously spiky summed calibrated radiocarbon proba- bility distributions. Even more remarkable is how often authors and reviewers do not recognize these spikes as critically serious anomalies (Fig. 4) The cause of calibration spikes The cause of the spikes is, however, easily under- stood using Fourier transform theory. Following the initial construction of the Gaussian on the 14C-scale (which has an inbuilt area=1 normalization), during (or following) 14C-calibration there is no need for further normalization. Under a Fourier transform there is only one function. Hence, once it is norma- lized, there is no reason to normalize the same func- tion, that is already normalized, a second time. The unnecessary application of secondary SPD-normali- zation is confirmed by Enrico R. Crema and Andrew Bevan (2020). Shape correction of 14C-histograms Many papers have been published with approaches that need either shape correction of archaeological and environmental 14C-histograms (e.g., Stolk et al. 1994), summed calibrated distributions (e.g., Wil- liams et al. 2012), or else – more recently – for their Bayesian counterparts in the form of kernel density plots (Bronk Ramsey 2017; Feeser et al. 2019; Lof- tus et al. 2019; Capuzzo et al. 2020; Mazzucco et al. 2020). An idea common to all these approaches is that since we can relate the existence of certain peaks, troughs, or spikes in the diagrams to the cal- ibration curve shape, we expect it should be possi- ble to apply an appropriate correction to the histo- Fig. 4. Blue curve: Summed Calibrated Probability Distribution (SCPD) for 5464 14C-ages from 1147 archaeological sites in South America. Red curve: same data with 400-year moving average. Both graphs are redrawn from Goldberg et al. (2016; ebda.Fig. 3b). The authors suggest that the recur- ring mid-Holocene peaks and troughs (9 ka to 5.5 ka) cannot be explained by calibration artefacts. We suspect that the spikes are not real. They are anomalies caused by secondary normalization. Bernhard Weninger, Kevan Edinborough 546 gram shape (on one or the other scale). Some re- searchers prefer just to get rid of the spike-anoma- lies by smoothing. However, when smoothing the spikes to get rid of them they are still there, albeit obscured. In contrast, and quite simply, because with a Fourier transform the spikes are not pro- duced, there is nothing that would require smooth- ing let alone correction. Even for the peaks and troughs many researchers are falling victim here to the misleading language developed long ago by ra- diocarbon specialists, who with good intention em- phasize (sometimes even today) that measured 14C- ages are older than expected by archaeologists, but which require age-corrections to allow for the secu- lar variations in atmospheric 14C-contents, and that these corrections can be achieved by a procedure called “14C-age calibration”. Unfortunately for the validity of this hypothesis, although fortunately for nature, nobody has yet demonstrated the feasibility of applying corrections to any of the many global physical processes that God (apparently with great wisdom) has undertaken great efforts so that hu- mankind cannot understand them, at least not im- mediately. The conceptual difficulty in our view, is not the actual histogram correction. The problem is escaping from the illusion that this normalization/ correction really is useful, even when achieved. Based on extensive modelling experiments, our pre- vious (and still valid) conclusion (Weninger et al. 2015) is that an (apparent) histogram shape correc- tion is indeed possible, although it only seemingly works correctly, and only then under very limited ideal modelling conditions, and that is for extreme- ly dense and exactly uniform sample distributions, and which only contain 14C-ages with exactly equal 14C-standard deviations. Yet, even under these ideal modelling conditions the actually achieved perfect correction of the 14C-histogram shape, under clos- er scrutiny it is nothing more than a chimera or ma- thematical anomaly, as it were; we see no reason that this should only apply to the non-Bayesian his- togram method. Instead – and this is what we now conclude – as a result of the Fourier transform there appears to exist some kind of fundamental mathe- matical rule that not only restricts, but actually for- bids the histogram shape correction. This restriction, in the language of quantum theory, surely has to do with the uncertainty relation (see below) and is maybe even indicative for what physicists call the second quantization. This is a concept first intro- duced in 1927 by Paul Dirac, in order to generalize the application of quantum theory from single-par- ticle to multi-particle systems. From the experiments described in Weninger et al. (2015), it looks as if – under the Fourier transform – we have yet to learn more about the statistical laws that large sets of 14C- ages apparently follow. For data sets that contain indistinguishable Bosons (i.e. large sets of 14C-ages with identical standard deviations; in the language of Fourier transform: waves with identical frequen- cy, such as lasers), new phenomena emerge that took physicists considerable time to figure out. As is today well known in quantum theory, the proper- ties of multi-particle systems that contain indistin- guishable particles (bosons: examples are magnets, lasers, supra-conductivity) can be very different from systems that contain distinguishable particles (fer- mions: examples are electrons forced to occupy dif- ferent atomic states under the Pauli exclusion prin- ciple). Under appropriate statistical conditions, as in 14C-analysis under the Fourier transform, we may therefore confidently expect many of the larger (multi-body) assemblages of 14C-bosons to have exactly the statistical properties as those described above, and which we unwittingly stumbled over, in our efforts to understand why it is so clearly impos- sible to find a correct histogram shape-correction. With this explanation we can replace the previous description, which Aristophanes may have termed “perfect cloud-cuckoo-land” (Hall, Geldart 1906. 820; boulei Nejelokokkugian) by a somewhat more technical description: we are looking at a sec- ond (deeper) level of quantization. Single Gaussian quantization What we presently know, at least, is that the impos- sibility of correct histogram shape-correction not only applies to larger sets of 14C-ages, but already to single 14C-ages. This forecasting (again under non- Bayesian conditions) may sound curious if not out- right wrong. Surely it is obvious that a single short- lived sample cannot possibly store the atmospheric 14C-content for any of the years before (or after) it was actually growing, and even more impossible, if we haven’t yet 14C-dated the sample? Indeed. Never- theless, we are in a quantum system. Therein, and whether we like it or not: 14C-ages follow the rules of the Fourier transform. The point being, we do not know the sample age. The direct (and famous) ana- logy to quantum physics for this forecasting would be that, within a double-slit experiment, the wave/ particle can show interference with itself. An opti- cal diffraction-pattern is observed, curiously, even when the intensity of the incoming particles is so strongly reduced that at any one moment there is only one single particle in the system. This particle, Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 547 we suppose, can only pass through one, or the other, of the two slits, but not both at the same time. There are many online illustrations of optical diffraction- patterns that are produced when single particles are allowed to build up an interference pattern on a screen, even though they arrive one by one. The Santorini dilemma As an alternative to reproducing one of the easily accessible Thomas Young’s interference graphs, in Figure 5 we have assembled from the literature some empirical data for the Santorini eruption. This may serve as an archaeologically more compelling illus- tration for the occurrence of wave diffraction pat- terns under a Fourier transform, and which also illu- strates Young’s interference. What first emerges is a picture that shows how com- plicated single-event 14C-dating can be, even under quasi-ideal research conditions. The unresolved ra- diometric problems for the Santorini dating include the possibility of interlaboratory offsets in the order of ±10 BP, in parallel to further issues concerning the existence of geographic and seasonal reservoir offsets, in the same order of magnitude (Manning et al. 2020). In terms of actually dating the Santorini eruption, as illustrated in Figure 5 the scales are even today still well balanced between the alterna- tive (high-middle-low) chronological hypotheses. This is not the place to advocate one or the other chronology. But it is interesting to see how the scales are now re-balancing in support of an intermediate date in the middle of the 16th century cal BC (Fan- tuzzi 2018; Pearson et al. 2020), or even younger. Looking at Figure 5 we can see that the now 10-year old statement by Malcolm H. Wiener (2009.203) is as true as ever: “Most radiocarbon measurements fall within the oscillating portion of the radiocar- bon curve, which makes it impossible to distin- guish dates between 1615 and 1525 BC”. A pathway to the solution is proposed below. From the methodological perspective, and foremost appa- rent, is that the error-simulated calibrated distribu- tions (corresponding to high-low shifted input 14C- Gaussians) shown in Figure 5 have properties that are similar to the above-mentioned optical single- particle diffraction-pattern. In principle, although we are comparing here a physical system (optics) with a mathematical structure (14C-calibration), both have the same underlying cause. Namely, when viewed from the perspective of a Fourier transform, both single-particle Young’s interference as well as single-date 14C-calibration can be mathematically de- scribed as transformations that decompose sharp/ compact input signals (Gaussians) into output sig- nals (dispersed waves) that have widely oscillating amplitudes at high frequencies Under such conditions, a suitable research concept would be to temporarily refrain from further efforts to obtain a direct 14C-based Santorini eruption date. It appears unlikely that the necessary natural scien- tific variables will be clarified in the near future (de- cadal scale). Pottery dating by correspondence analysis A promising solution to the Santorini Dilemma, from an archaeological perspective, would be to expand on the already now available dating preci- sion of c. ±20 yrs (95% confidence). This is not any futuristic dating precision, but the starting precision for statistical seriation of Mycenaean pottery found in Helladic and Minoan deposits. As demonstrated in Figure 6 for the pottery data published by Arne Fu- rumark (1972a,b), this dating precision is an order of magnitude (factor 10) better than achieved by single-particle 14C-Fourier analysis at Santorini. Naturally, Furumark’s classification needs much up- dating and geographic extension in Helladic and Mi- noan pottery studies. His study was completed in 1940 (Furumark 1972a.15). It is 80 years old. Hence, when updated, the precision achieved with the CA is likely to increase (we hypothesize). Let’s put it another way. Santorini is one site, but one which has attracted and focussed considerable at- tention for quite some long time. Of course, it is an important site. Yet, by CA-application, it is possible – based on an updated pottery database and clas- sification that can be readily constructed from the literature – to provide a large number of archaeolo- gical sites with high-precision pottery dates, simul- taneously, for many regions of the eastern Mediter- ranean. Furthermore, if this work is initiated, we may forecast the discovery of many more Santorini- type dating discrepancies than are presently known. Put differently, we forecast that the now well-stud- ied Santorini Dilemma will be widely observable, and similarly CA-resolvable, at other sites in the eastern Mediterranean. As an important component of this dating program, there will be need for the critical combination of large numbers of 14C-mea- surements and historical dates. There will be little need, however, for further distraction of the archa- eological research by localized dating controversies. Bernhard Weninger, Kevan Edinborough 548 The distinction between classical and Bayesian theory Another rather knotty problem of the assumed 14C- foundation in Bayes’ theorem pertains to the dis- tinction, also introduced by Dehling and van der Plicht (1993), that we should differentiate between classical and Bayesian approaches to statistical the- ory. This is easier to understand than the difference between a probability and likelihood ratio (function). In this case it is quite simply the terminology used by Dehling and van der Plicht (1993) that contrasts with much of the language used in physics. In phy- sics the classical version of a theory is always the one that uses commutative variables (and which re- minds us of Isaac Newton), whereby quantum theo- ry supports the analysis of noncommutative vari- ables (esteemed names like Werner Heisenberg, Paul Dirac, and John Neumann spring to mind). We will return to Heisenberg’s uncertainty principle below. Fig. 5. Calibration of the 14C-scale weighted average 3345±8 BP for the Minoan eruption (Akrotiri Volca- nic Destruction Level) from Manning et al. (2014) (blue Gaussians) and two shifted (high-low) dates in the range of 20 BP (grey Gaussians) against INTCAL20 (Reimer et al. 2020). Graphic-overlay of annual tree-ring data from Pearson et al. (2018; 2020). Note: (1) The Arizona-lab data (green bars) by Pearson et al. (2018) is included in INTCAL20 construction, but not (red bars) the data by Pearson et al. (2020); (2) INTCAL20 is not only based on Arizona lab data (cf. Reimer et al. 2020; Heaton et al. 2020). Due to unresolved interlab variability, unknown growth-season, and possible geographic carbon-cycle differen- ces, each in the range ±10 BP, this comparison indicates that even ±8 BP dating precision is not suffi- cient at the present stage of research to gain a final solution on the Minoan eruption dating. Ultimately, this is due to unknown error propagation in a Fourier transform that is very sensitive to high-frequen- cy signal fluctuation. A good analogue for the Santorini dating dilemma, next to optical Young interfe- rence, is a faulty electrical connection (in German Wackelkontakt). Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 549 In quantum physics, it is the uncertainty principle in particular that is associated with practically all known quantum properties of atomic and nuclear states, and of elementary particles, such that it is un- derstood as responsible (generic) for all observed quantization effects. From a mathematical perspec- tive, in 14C-analysis just as in quantum physics, the uncertainty principle follows directly from the ma- thematical properties of the Fourier transform. The noncommutativity of 14C-ages In analogy to the double-slit experiment, Figure 7 shows the 14C-Fourier transform of a Gaussian-shap- ed 14C-measurement. This figure recalls that when 14C-ages are calibrated, different results are achieved, depending on whether the mathematical operation of averaging is performed on the 14C-scale (before calibration) or calendric scale (after calibration). In mathematical terminology, this property of the 14C- calibration operator is known as noncommutative. From the description of 14C-calibration as a Fourier transform, it follows naturally that one and the same mathematical function can have different appear- ances, depending on the domain from which it is visualized. This is illustrated in Figure 7 (A-D) for ini- tially only one Gaussian 14C-measurement, called A, but with later repetition B for the same sample (a=b) and with A=B [BP]. The comparison of Figure 7 (A,B,C, D) illustrates that even when we are truly desperate in a chronolo- gical study to achieve highest pos- sible (calibrated) dating precision, it seldom helps to repeat the 14C- measurement on the same sample. This is because the repeat measu- rement (performed on the 14C-do- main) does not provide any signi- ficant enhancement of the calibrat- ed sample age (viewed on the ca- lendric domain), even under ideal conditions. From the perspective of the Fourier transform, all that is achieved by repeating the measu- rement on the 14C-scale is to re- place an already existing particle with its identical copy. A humorous illustration that de- monstrates how deeply the Fourier transform is embedded within quantum theory is derived from the so-called One- Electron Universe hypothesis of John Wheeler and Richard Feynman. According to the story told by Jagdis Mehra, in his splendid description of Richard Feynman’s scientif- ic and other achievements, it so happened that: “... at about the same time, in the fall of 1940, Feyn- man received a telephone call from John Wheeler at the Graduate College in Princeton, in which he said that he knew why all electrons have the same charge and the same mass. ‘Why?’ asked Feynman, and Wheeler replied, ‘Because they are all one and the same electron.” (Mehra 1996.113) This so-called One-Electron Universe is illustrated in Figure 8.A-B. It shows three wobbly lines drawn to represent the individual world-lines of three dif- ferent electrons, called Particles 1, 2 and 3. But now, as shown in Figure 8.C, after turning the graph by 90˚ the same graph (with minor changes) has every appearance of the 14C-age calibration curve, such that, even with only one 14C-Gaussian to be cali- brated, this one Gaussian may have any number of different calibrated ages. Note, however, that the analogy between the One-Electron-Joke (OEJ), the Calibration Curve (CC) and the Fourier Transform Fig. 6. Seriation of Furumark’s pottery/decoration types by corres- pondence analysis. The horseshoe shape of the distribution allows to establish the relative order and relative chronology with a preci- sion (1st order estimate) of ±20 yrs (95%-confidence), often better. Data: Furumark (1972). Note: the estimated dating precision is va- lidated for excavation units (not shown) from Kalapodi, Lefkandi, and Mycenae (Granary). Bernhard Weninger, Kevan Edinborough 550 (FT) is only valid for the CC-FT comparison, but strict- ly speaking not for OEJ-FT, for scaling reasons given in the Appendix Nr: [7]. First quantization properties (single and group- ed 14C-dates) Such properties of 14C-dates we call first quantiza- tion, and these are omnipresent in archaeological 14C-analysis. They have an amplitude far beyond the statistical noise of the 14C-measurements. The first quantization properties of 14C-dates can be clas- sified according to their occurrence for single dates, data groups, and data series. Single 14C-dates ● Lock-in of numeric age-values for confidence in- tervals (e.g., 95% or 68%) that are used to abbre- viate calendric-scale age distributions, also for mul- tiple disjunct intervals ● Separation of calendric-scale confidence intervals into multiple disjunct regions ● Dispersal of the calibrated Gaussian and its sepa- ration into different components on the calendric timescale ● Lateral shift of the calibrated median along the ca- lendric time-scale ● Dispersal and lateral shift of the area normalized Gaussian on the 14C-scale ● Probability values assigned to multiple disjunct intervals seldom sum to 100%. Data groups For larger sets of radiocarbon ages (Data Groups) the properties assigned to the individual 14C-ages are all similarly observable, but combine to produce the following new quantization effects: ● clustering of 14C-ages on the 14C-scale; ● clustering of readings on the calendric time-scale; ● attraction of 14C-ages towards predefined inter- vals on the 14C-scale; ● attraction of calendric readings toward as prede- fined intervals on the calendric scale. Fig. 7. Illustration of the noncommutative properties of 14C-calibration and construction of gauge-pla- teaus for (input) single 14C-domain Gaussian and (output) double-reading calendric domain quasi-Gaus- sians. Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 551 Second quantization properties (grouped 14C- data) For archaeologically sequenced or otherwise tempo- rally seriated data sets (Structured Data Groups) all quantization effects noted above for Single 14C-dates and data groups are known to occur, but under cer- tain conditions some new effects – we call second quantization – can be observed. Above, we have al- ready attributed the non-correctability of the histo- gram shape to so-called second quantization effects. As a reminder, these become observable for large assemblages of indistinguishable particles, in our case for large 14C-data sets of closely packed and equidistant samples, best observable when all 14C- measurements have equal standard deviations. We then have sets of bosons. Second quantization properties (seriated 14C- data) Quite generally, the second (boson-analog) quantiza- tion is more difficult to recognize than the First, but not because it is weaker. Simply, it occurs mainly for larger data sets, and under certain conditions, but which are relatively rare. By theoretical considera- tion, we would expect such effects not only to occur for weakly coupled (i.e. grouped) 14C-data, but to be even more visible for more strongly coupled (i.e. se- riated) data, such as tree-ring sequences. In such cases, indeed, the pre-established sample order can be so strong as to completely prevent (inhibit) any changes in the structure of the data set. Nonetheless, since the quantization effects cannot be turned off, they are still operative. What happens? Radiocarbon quantization: frozen data struc- tures Take by way of example a tree-ring sequence of 14C- ages with precisely measured (say error-free) calen- dric-scale distances between the samples. In such a data set the sample order is so tightly restricted that the internal structure of the data set is – so to say – frozen. In this frozen state, when fitted to the cali- bration curve the remaining (only possible) quanti- zation reaction is to increase the number of best-fit calendric ages en bloc for the entire data set. In con- sequence, the only remaining reaction is that the wiggle-matching will show multiple (logically alter- native) best-fit solutions. We have observed this ef- fect, and what we now call Frozen Data Structures, in many wiggle-matching studies. Although present- ly without formal proof, the occurrence of such fro- zen subset components is what we presently hypo- thesize has caused block-wise age distortion in Baye- sian sequencing at Assiros (North Greece, Late Bronze Age), in this case for a mixture of strongly coupled Fig. 8. A-B the One-Electron Universe hypothesis of Wheeler and Feynman. Redrawn from Mehra (1996. Fig.5.1, 5.2); C comparison with 14C-Age Calibration System. Bernhard Weninger, Kevan Edinborough 552 tree-ring sequences and weakly coupled bone data (Gimatzidis, Weninger 2020.ebda.Fig. 2). Nonethe- less, we acknowledge, such properties of sequenced 14C-data sets are presently at the limit of visibility. Heisenberg’s uncertainty principle Given that even the most precisely fitting 14C-data sets (e.g., tree-ring sequences) show such strong ca- libration lock-in effects, we may expect that – ulti- mately – there must exist a generally applicable ma- thematical theorem that neatly forecasts all observ- able quantization properties of 14C-data, whether for single dates, data groups, or data series. For the mo- ment, we do not know how to formulate this theo- rem, but there are good chances that it will look si- milar to the Fourier transform uncertainty relation, shown in Equation 2. A shorter version of the same equation is found in many textbooks of quantum physics, where it is adapted to the properties of wave/particles and known as Heisenberg’s uncer- tainty principle (Eq. 3). The statement in common to both equations is that efforts to sharpen the study function in one domain will inevitably lead to its spread in the second do- main. Therein we at last have a satisfactory mathe- matical explanation for the curious, and for dating experts the counter-intuitive observation that under Bayesian sequencing one may indeed achieve higher dating precision, but only at the loss of dating accu- racy (Steier, Rom 2000). Please note that reason- ably understandable surveys of the mathematical connections between the Fourier transform and the uncertainty principle are already provided by Gerald B. Folland and Alladi Sitaram (1997) and Dhiman Sen (2014). To these connections we may now add the process of 14C-calibration. Discussion In the title of the paper our assertion is that the Fou- rier transform represents a beautiful foundation for 14C-calibration. Yes, there may be concerns about the validity of our beautiful calibration hypothesis, which may in consequence produce some immedi- ately critical debate. Whether for mathematical, phy- sical, epistemological, or aesthetic reason, this is of no consequence, if our learned readers allow us to resolve one last question. That is: why do you de- scribe such an apparently trivial, and certainly tech- nical proposal as beautiful? The answer is four-fold. First, we attribute the beauty of 14C-calibration to the sublime mathematical symmetry of the underly- ing Fourier transform; second, to the many remark- able analogies between 14C-calibration and quan- tum theory, thirdly, to the unprecedented explana- tory usefulness of Fourier-based 14C-calibration, as well as – finally – to the curiously nonconformist yet immediately understandable appearance of the Fou- rier transform in an unexpected context. Perhaps our Fourier transform hypothesis is, in itself, not strikingly beautiful, but it is undeniably elegant. Conclusion In this paper we propose a rethinking of the mathe- matical foundation of archaeological 14C-age calibra- tion. We also suggest that archaeologists have at least as much to learn from physicists as they do from mathematicians and statisticians. Following many years of dedicated education, persistent tech- nical support, and admirable instruction by radiocar- bon dating experts, parts of the archaeological com- munity are close to the erroneous conclusion that procedures underlying 14C-calibration follow directly from Bayesian probability theory. The choice of a Bayesian framework in 14C-analysis offers, indeed, highly luxurious analytical conditions for archaeo- logical age-modelling. Next to established luxury and acclaimed beauty, the process of 14C-calibration is better described as the Fourier transform. ( ( ) ) ( ( ) )x f x dx f d2 2 2 2 2 1 16 ⋅ ≥ −∞ +∞ −∞ +∞ ∫ ∫ ξ ξ ξ π [Eq. 2] Δ Δx p h≥ 2 [Eq. 3] We thankfully acknowledge many years of support and motivation by Andy Bevan (London), Lee Clare (Berlin), Tiziano Fantuzzi (Venice), Olaf Jöris (Mon- repos), Raiko Krauß (Tübingen) and Reinhard Jung (Wien). We also thank an anonymous reviewer whose insightful comments certainly improved this manu- script. ACKNOWLEDGEMENTS Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 553 Andersen K. K. and 12 co-authors. 2006. The Greenland Ice Core Chronology 2005, 15–42 ka. Part 1: constructing the time scale. Quaternary Science Reviews 25: 3246– 3257. https://doi.org/10.1016/j.quascirev.2006.08.002 Bayliss A. 2009. Rolling out revolution: Using Radiocar- bon Dating in Archaeology. Radiocarbon 51(1): 123– 147. https://doi.org/10.1017/S0033822200033750 Bell A. E. 1942. Hypotheses Non Fingo. Nature 28(149): 238–240. Blaauw M., Christen J. A. 2011. Flexible paleoclimate age- depth models using an autoregressive gamma process. Bayesian Analysis 6: 457–474. https://chrono.qub.ac.uk/ blaauw/bacon.html. Visited June, 2020. Bronk Ramsey C. 2009. Bayesian Analysis of Radiocarbon Dates. Radiocarbon 51(1): 337–360. https://doi.org/10.1017/S0033822200033865 2017. Methods for Summarizing Radiocarbon Datasets. Radiocarbon 59(6): 1809–1833. https://doi.org/10.1017/RDC.2017.108 2020. OxCal Calibration Software. https://c14.arch.ox.ac.uk/oxcal.html Buck C. E., Meson B. 2015. On being a good Bayesian. World Archaeology 47(4): 567–584. https://doi.org/10.1080/00438243.2015.1053977 Buck C. E., Christen J. A., and James G. N. 2020. An on- line Bayesian Calibration Tool. https://bcal.shef.ac.uk/ Buck C. E., Juarez M. 2017. Bayesian radiocarbon model- ling for beginners. arXiv:1704.07141v1. [stat.AP]: 1–26. Buck C. E., Kenworthy J. B., Litton C. D., and Smith A. F. M. 1991. Combining archaeological and radiocarbon infor- mation: a Bayesian approach to calibration. Antiquity 65: 808–821. https://doi.org/10.1017/S0003598X00080534 Buck C. E., Litton C. D., and Smith A. F. M. 1992. Calibra- tion of radiocarbon results pertaining to related archaeo- logical results. Journal of Archaeological Science 19: 497–512. Buck C. E., Meson B. 2015. On being a good Bayesian. World Archaeology 47(4): 567–584. https://doi.org/10.1080/00438243.2015.1053977 Capuzzo G. and 15 co-authors. 2020. Cremation vs. Inhu- mation: Modelling Cultural Changes in Funerary Practises from the Mesolithic to the Middle Ages in Belgium using Kernel Density Analysis on 14C-Data. Radiocarbon: Sep- tember 2020. https://doi.org/10.1017/RDC.2020.88 Crema E. R., Bevan A. 2020. Inferences from large sets of Radiocarbon dates: software and methods. Radiocarbon: October 2020. https://doi.org/10.17863/CAM.55924 Dehling H., van der Plicht J. 1993. Statistical Problems in Calibrating Radiocarbon Dates. Radiocarbon 35(1): 239– 244. https://doi.org/10.1017/S0033822200013928 Douka K., Laurent C., Nespoulet R., and Higham T. 2020. A refined chronology for the Gravettian sequence of Abri Pataud. Journal of Human Evolution 141: 102730. https://doi.org/10.1016/j.jhevol.2019.102730 Einstein A. 1915. Erklärung der Perihelbewegungen des Merkur aus der Allgemeinen Relativitätstheorie. Sitzungs- berichte der Preußischen Akademie der Wissenschaften: 831–839. (Referenced according to von Mayenn, 1990, Abhandlung [7], 234–246). Fantuzzi T. 2018. A Reassessment of the Debate onm Late Minoan I and interlinked Chronologies through Radiocarbon and Comparative Analysis. PhD Thesis. Ca’Foscari University of Venice. Venice. Feeser I., Dörfler W., Kneisel J., Hinz M., and Dreibrodt S. 2019. Human impact and population dynamics in the Neo- lithic and Bronze Age: Multi-proxy evidence from north- western Central Europe. The Holocene 29(10): 1596– 1606. https://doi.org/10.1177/0959683619857223 Feynman R. 2010. The Feynman Lectures on Physics. Vo- lume III: Quantum Mechanics. Basic Books. New York Fließbach T., Walliser H. 2012. Arbeitsbuch zur Theore- tischen Physik. Spektrum. Akademischer Verlag Heidel- berg. Heidelberg. Folland G. B., Sitaram A. 1997. The Uncertainty Principle: A Mathematical Survey. The Journal of Fourier Analysis and Applications 3(3): 207–238. Furumark A. 1972a. Mycenaean Pottery I. Analysis and Classification. Skrifter Utgiva AV Svensks Institutet I Athen. Acta Institutu Atheniensis Regni Sueciae 4, XX:1. Stockholm. 1972b. Mycenaean Pottery II. Chronology. Skrifter Ut- giva AV Svensks Institutet I Athen. Acta Institutu Athe- niensis Regni Sueciae. 4, XX:2. Stockholm. Geyh M. A. 1969. Versuch einer chronologischen Gliede- rung des marinen Holozäns an der Nordseeküste mit Hilfe References Bernhard Weninger, Kevan Edinborough der statistischen Auswertung von 14C-Daten. Zeitschrift der Deutschen Geologischen Gesellschaft 118: 351–360. Gimatzidis S., Weninger B. 2020. Radiocarbon dating the Greek Protogeometric and Geometric periods: The evi- dence of Sindos. PLoS ONE 15(5): e0232906. https://doi.org/10.1371/journal.pone.0232906 Grootes P. M., Stuiver M., White J. W., Johnsen S., and Jou- zel J. 1993. Comparison of Oxygen Isotope Records from the GISP2 and GRIP Greenland Ice Core. Nature 366: 552–554. https://doi.org/10.1038/366552a0 Goldberg A., Mychajaliw A. M., and Hadly E. A. 2016. Post- invasion demography of prehistoric humans in South America. Nature 532: 232–235. https://doi.org/10.1038/nature17176 Hall F. W., Geldart W. M. 1906. Aristophanes Comoediae. Volume 1. Clarendon. Oxford. Heaton T. J., Blaauw M., Blackwell P. G., Bronk Ramsey C., Reimer P. J., and Scott E. M. 2020. The INTCAL20 ap- proach to Radiocarbon Calibration Curve Construction: A new methodology using Bayesian Splines and Errors-In- Variable. Radiocarbon 62(4): 821–863. https://doi.org/10.1017/RDC.2020.46 Hermann R. 1970. Lectures in Mathematical Physics. Vo- lume I. W. A. Benjamin, Inc. New York. Katsianis M., Bevan A., Styliaras G., and Maniatis Y. 2020. An Aegean History and Archaeology Written through Ra- diocarbon Dates. Journal of Open Archaeology Data 8: 5. https://doi.org/10.5334/joad.65 Loftus E., Mitchell P. J., and Bronk Ramsey C. 2019. An archaeological radiocarbon database for southern Africa. Antiquity 93(370): 870–885. https://doi.org/10.15184/aqy.2019.75 Lutomski M. G. 2013. The Use of Quantitative Risk Asses- sment in the Operations Phase of Space Missions. In T. Sgobba, A. F. Allahdadi, I. Rongier, and P. D. Wilde (eds.), Safety Design for Space Operations. Butterworth-Heine- mann, Elsevier. Amsterdam: 805–828. Manning S. W., Höflmayer F., Moeller N., Dee M. D., Bronk Ramsey C., Fleitmann D., Higham T., Kutschera W., and Wild E. M. 2014. Dating the Thera (Santorini) eruption: archaeological and scientific evidence supporting a high chronology. Antiquity 88(342): 1164–1179. https://doi.org/10.1017/S0003598X00115388 Manning S. W., Kromer B., Cremaschi M., Dee M. W., Frie- drich R., Griggs C., and Gadden C. S. 2020. Mediterranean radiocarbon offsets and calendar dates for prehistory. Sci- ence Advances 6: eaaz1096. DOI: 10.1126/sciadv.aaz1096 Mayewski P. A., Meeker L. D., Twickler M. S., Whitlow S. I., Yang Q., Lyons W. B., and Prentice M. 1997. Major fea- tures and forcing of high-latitude northern hemisphere at- mospheric circulation using a 110,000-year-long glacio- chemical series. Journal of Geophysical Research 102: 26345–26366. https://doi.org/10.1029/96JC03365 Mazzucco N., Ibáñez J. J., Capuzzo G., Gassin B., Mineo M., and Gibaja J. F. 2020. Migration, adaptation, innovation: The Spread of Neolithic harvesting technologies in the Mediterranean. PLoS ONE 15(4): e0232455. https://doi.org/10.1371/journal.pone.0232455 Mehra J. 1996. The Beat of a Different Drum. The Life and Science of Richard Feynman. Oxford University Press. Oxford. Messiah A. 1976. Quantenmechanik, Band I. Walter de Gruyter, Berlin, New York. Nicholls G., Jones M. 2000. Radiocarbon dating with temporal order constraints. Technical Report #407. Ma- thematics Department. University of Auckland. April 1998, revised September 2000. Auckland: 1–20. Milner S. 2018. Newton didn’t frame hypotheses. Why should we? Physics Today, 24 Apr 2018 (Commentaries & Reviews). https://doi.org/10.1063/PT.6.3.20180424a Pearson C. L., Brewer P. W., Brown D., Heaton T. J., Hod- gins G. W., Jull A. T., Lange T., and Salzer M. W. 2018. An- nual radiocarbon record indicates 16th century BCE date for the Thera eruption. Science advances 4(8): eaar8241. https://doi.org/10.1126/sciadv.aar8241 Pearson C., Salze, M., Wacker L., Brewer P., Sookdeo A., and Kuniholm P. 2020. Securing timelines in the ancient Mediterranean using multiproxy annual tree-ring data. Proceedings of the National Academy of Sciences, 117 (15): 8410–8415. https://doi.org/10.1073/pnas.1917445117 Pettitt. P., Zilhão J. (eds.) 2015. Problematizing Bayesian approaches to prehistoric chronologies. World Archaeo- logy 47(4): 525–542. https://doi.org/10.1080/00438243.2015.1070082 Reimer P. J. and 41 co-authors. 2020. The INTCAL20 Northern Hemisphere Radiocarbon Age Calibration Curve (0–55 Cal kBP). Radiocarbon 62(4): 725–757. https://doi.org/10.1017/RDC.2020.41 Rohling E. J., Marino G., Grant K. M. L., Mayewski P. A., and Weninger B. 2019. A model for archaeologically rel- evant Holocene climate impacts in the Aegean-Levantine region (easternmost Mediterranean). Quaternary Science Reviews 208: 38–53. https://doi.org/10.1016/j.quascirev.2019.02.009 554 Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 555 Schenk W., Kremer F., Beddies G., Franke Th., Galvosas P., and Rieger P. 2014. Physikalisches Praktikum. 14. Auflage. Springer Spektrum. Wiesbaden Schmiedl G., Kuhnt T., Ehrmann W., Emeis K.-C., Hamann Y., Kotthoff U., Dulski O., and Pross. J. 2010. Climatic forc- ing of eastern Mediterranean deep-water formation and benthic ecosystems during the past 22 000 years. Quater- nary Science Reviews 29(23–24): 3006–3020. https://doi.org/10.1016/j.quascirev.2010.07.002 Sen D. 2014. The uncertainty relations in quantum me- chanics. Current Science 107(2): 203–218. Sharafi A., Pourmand A., Canuel E. A., Ferer-Tyler E., Pe- terson L. C., Aichner B., Feakins S. J., Daryaee T., Djamali M., Naderi Beni A., Lahijani H. A. K., and Swart P. K. 2015. Abrupt climate variability since the last deglaciation based on a high-resolution, multi-proxy peat record from NW Iran: The hand that rocked the Cradle of Civilization. Quaternary Science Reviews 123: 215–230. https://doi.org/10.1016/j.quascirev.2015.07.006 Skilling J., Knuth K. H. 2019. The Symmetrical Foundation of Measure, Probability, and Quantum Theories. Annalen der Physik 531: 1–9. https://doi.org/10.1002/andp.201800057 Steier P., Rom W. 2000. The use of Bayesian statistics for 14C dates of chronologically ordered samples: a critical analysis. Radiocarbon 42(2): 183–98. https://doi.org/10.1017/S0033822200058999 Stolk A., Törnqvist T. E., Hekhuis K. P. V., Berendsen H. J. A., and Van der Plicht. J. 1994. Calibration of 14C Histo- grams: A Comparison of Methods. Radiocarbon 36(1): 1– 10. https://doi.org/10.1017/S0033822200014272 Stuiver M., Reimer P. J., and Reimer R. W. 2020. CALIB 7.1. CALIB Radiocarbon Calibration. http://calib.org, ac- cessed 2020-6-17. van Hove L. 1958. Von Neumann’s contributions to Quan- tum theory. Bulletin of the American Mathematical So- ciety 6(3): 95–99. von Meyenn K. 1990. (ed.) Albert Einsteins Relativitäts- theorie. Die grundlegenden Arbeiten. Friedr. Vieweg & Sohn Verlagsgesellschaft mbH. Braunschweig. von Neumann J. 1927. Mathematische Begründung der Quantenmechanik. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikali- sche Klasse: 1–57. https://eudml.org/doc/59215 Weninger B. 1986. High-precision calibration of archaeo- logical radiocarbon dates. Acta Interdisciplinaria Archaeo- logica, Tomus IV, 11–53. In E. Neustupný (ed.), Papers of the Symposium held at the Institute of Archaeology of the Sloval Academy of Sciences, Nové Vozokany, Oc- tober 28–31, 1985. Archaeologický ústav Slovenskej aka- démie vied. Nitra: 11–54. Weninger B. and 18 co-authors. 2009. The Impact of Ra- pid Climate Change on prehistoric societies during the Ho- locene in the Eastern Mediterranean. Documenta Praehi- storica 36: 7–59. https://doi.org/10.4312/dp.36.2 Weninger B., Edinborough K., Clare L., and Jöris L. 2011. Concepts of Probability in Radiocarbon Analysis. Docu- menta Praehistorica 38: 1–20. https://doi.org/10.4312/dp.38.2 Weninger B., Clare L., Jöris O., Jung R., and Edinborough K. 2015. Quantum theory of radiocarbon calibration. World Archaeology 47(4): 1–24. https://doi.org/10.1080/00438243.2015.1064022 Weninger B. 2020. Barcode seriation and concepts of Gauge Theory. The 14C-Chronology of Star≠evo, LBK, and early Vin≠a. Quaternary International: In Press, Cor- rected Proof. Available online 25 April 2020. https://doi.org/10.1016/j.quaint.2020.04.031 Weninger F. 2011. Bayesian sequencing of radiocarbon dates. Unpublished PhD Thesis. Faculty of Physics Univer- sity of Vienna. Vienna. http://othes.univie.ac.at/15273/ 1/2011-06-06_8501664.pdf Weninger F., Steier P., Kutschera W., and Wild E. M. 2000. Robust Bayesian Analysis, an attempt to improve Baye- sian Sequencing. Radiocarbon 52(2–3): 962–983. https://doi.org/10.1017/S0033822200046075 Wichmann E. H. 1971. Quantum Physics. Berkeley Phy- sics Course – Volume 4. Mcgraw-Hill Book Company. New York. Wiener M. 2009. The State of the Debate about the Date of the Theran Eruption. In D. A. Warburton (ed.), Warbur- ton Time’s Up: Dating the Minoan Eruption of Santori- ni. Acts of the Minoan Eruption Chronology Workshop. Sandbjerg. November 2007. Monographs of the Danish Institute at Athens Volume 10. Athens: 197–206. Williams N. A. 2012. The use of summed radiocarbon pro- bability distributions in archaeology: a review of meth- ods. Journal of Archaeological Science 39(3):578–89. https://doi.org/10.1016/j.jas.2011.07.014 Winkler R. L. 1996. Uncertainty in probabilistic risk asses- sment. Reliability Engineering & System Safety 54(2–3): 127–132. [1] Translation of variables The most important translation rule is that, whenever in the Fourier transform an angular frequency (sym- bol w) or a wave frequency (symbol n) is involved, under 14C-analysis we translate the corresponding va- riable as 14C-age, measured on the [BP]-scale. Schematically this means: radiocarbon translates to Fourier transform 14C-Age m ± s [BP] <=> frequency n ± sn [sec–1] 14C-Age m ± s [BP] <=> angular frequency w ± sw [sec–1] In consequence, under the Fourier transform we must change the dimension of 14C-ages from [BP] to [1/sec]. This is only required as a Gedankenexperiment. At first sight it may seem curious if not outrightly wrong to see this translation of the well-known [‘14C-yrs’] dimension into an inverse time-scale: [sec–1]. However, from the perspective of archaeological 14C-dating this change in scaling is easily possible. It follows math- ematically from the needs of the Fourier transform. [2] Change of 14C-scale under the Fourier transform We emphasize that the change of 14C-scale from [BP] to [sec–1] necessary for the Fourier transform is a Ge- dankenexperiment. Traditionally, the Libby equation is used to define 14C-ages. However, Libby ages are measured, and provided to the user, according to the technical needs of 14C-measurement, which are diffe- rent from the mathematical needs of archaeological 14C-analysis. The required change in 14C-scale can be motivated by the following. The physical dimension of all 14C-dates is related to the amount of 14C remaining in the sample after its separation from the atmospheric carbon reservoir. From the perspective of Fourier transform, however, since the amount of 14C actually measured today in the sample (by whatever technique e.g., beta-decay, 14C-AMS, any other), is the prime reference value, and this value has the physical dimen- sions of [counts/sec], abbreviated [sec–1], it follows that 14C-measurements could – in principle – be given on the scale [sec–1]. From an archaeological perspective the actual [BP]-scale used is historically motivated but is otherwise secondary to the needs and requirements of 14C-dating (in contrast to 14C-measurement). [3] Notes on the mathematics of the Fourier transform Applications of the Fourier transform are typically based on complex-valued functions, for technical reasons. Computations with complex functions are easier to handle than sines/cosines. Even then, Fourier transform equations are often so complicated that they do not support a symbolic solution, but require numerical ap- proximation. However, there is one major exception, famously known to students, which is fundamental to our introduction of 14C-calibration as a Fourier transform. This engineering rule is: the Fourier transform of a Gaussian is again a Gaussian. Furthermore, another rule is important in 14C-analysis, namely: the Fourier transform is a linear operation. The Fourier transform of the sum of two functions is the sum of the trans- forms. [4] Why is the Fourier transform fundamental to 14C-age calibration? Together, these two rules guarantee the applicability of the Fourier transform to the 14C-histogram method, as introduced by Mebus A. Geyh (1969), and the calibration of 14C-histograms, according to the algorithm of Bernhardt Weninger (1986). Given that 14C-scale histogram construction is additive for measured Gaussians, and that the histogram is most easily and efficiently (inversely) calibrated by applying that algorithm (and which supports further Euclidian error analysis), then these two methods in combination constitute all that is needed for calibration of 14C-ages. It follows that, although useful as an approximation, there is no mathe- matical necessity for the foundation of 14C-calibration in Bayes’ theorem. [5] Axiomatic foundation of 14C-calibration? There have been ruminations in earlier radiocarbon literature that – in contrast to Bayesian 14C-calibration – the method of histogram-calibration has no foundation in probability theory. Such statements suggest that the respective authors do not acknowledge the existence of non-classical probability and measure theory. As is well-known in both theoretical and experimental physics, all modern quantum theory – of which the Fou- rier transform (and then in consequence 14C-calibration) constitutes an important component – has its ac- Bernhard Weninger, Kevan Edinborough 556 Appendix> 14C-age calibration under the Fourier transform Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 557 cepted and to some large extent axiomatic foundation in Hilbert Space theory. This quite remarkable result was achieved within the very short time span of two years (1927–1929) by John von Neumann (cf. van Hove 1958). The earliest version of his studies was published in 1927 in German (von Neumann, 1927). [6] What makes 14C-calibration a Fourier transform (FT)? A brief but complete answer is that, as in every Fourier transform, in 14C-analysis we have two scales (14C and calendric), between which the function under study can switch backwards and forwards. Accordingly, two formulas are needed to describe the 14C-calibration. Most typically, in archaeology the task is to calibrate a Gaussian shaped 14C-age. The basic equation for calibration of a Gaussian 14C-age m ± sm [BP} with medi- an m and standard deviation sm on a calibration curve r(t,t) for calendar ages with values t [cal BP] and 14C- scale with values t [BP] is given in Equation 1. As shown in Equation 2, the measured 14C-scale Gaussian ĝ(t) is area-normalized to unit 1. This supports interpretation of the underlying 14C-measurements as (ob- servable) probabilities. This interpretation is important in the laboratory for purposes of quality control and refinement of equipment and methods. The question of how to define (observable) probabilities for cali- brated 14C-ages, for similar purposes in archaeological chronology, is addressed below. t = radiocarbom scale [BP] [Eq. 1] Gaussian distribution 14C-Age: m ± sm [BP] 14C-measurement r(t,t) ± s(t,t) = calibration curve [a,BP] [BP] [Eq. 2] single Gaussian area normalisation (radiocarbon domain normalisation) The Fourier transform is widely applied in quantum physics (e.g., Feynman 2010; Fließbach, Walliser 2012; Hermann 1970; Messiah 1976; Schenk et al. 2014; Wichmann 1971). It is typically based on two paired equations. In Equation 3 and Equation 4 we have chosen to show the most often used Fourier equations, with variables adapted to archaeological 14C-analysis. In archaeology, we would traditionally be naming the (output) function g(t) as calibrated 14C-age [cal BP]. We might also say that the calibrated age g(t) belongs to the conventional 14C-age ĝ(t) [BP]. Such expression of ownership (belongs) conforms with the fact that, under the Fourier transform, there only exists one function, although it looks different in the two domains. Also important is that, in agreement with the International System of Base Units (SI), in Equations 1–4 we have defined the function ĝ(t) on the frequency axis t with physical units [sec–1], in replace of 14C-yrs [BP]. Its twin function g(t) is defined on the time axis t with physical units [sec], in replace of calendric years [a]. The standard radiocarbon layout for the paired Fourier transform equations is then as follows: [Eq. 3] inverse direction Fourier transform paired equations [Eq. 4] forward direction [7] Scales with complementary dimensions For the Fourier transform to work properly, the functions ĝ(t) and g(t) – as defined on the two domains (14C and cal-scale) – must have complementary dimensions. This guarantees that their product ĝ(t)·g(t) has unit = [1] dimension. For example (as chosen here), if time t is measured in seconds [sec], then the same function but shown in the second domain must be measured using the inverted unit [sec–1]. That is, for radioactivity measurements, the scale of counts per sec. When measuring frequencies, the correspond- ing dimension would be cycles per second. For the Fourier transform the only important thing is that the product of the two dimensions is equal to unit=1. When the Fourier transform is applied to 14C-analysis, the calendric time scale automatically has the correct time-dimension of [sec], with trivial rescaling of calendar years to seconds. Hence – after some thought – it becomes clear that under the Fourier transform it is only the 14C-scale that requires non-trivial rescaling (as mentioned above), from [BP] to [sec–1]. The details of this g( ) = 1 2 e 1 2 r(t, τ σ π μ τ σ ( ) ) − 2 σ σ σ τμ= + 2 2 ( ) g( )dτ τ = −∞ +∞ ∫ 1 ˆ ˆ g( ) g(t)e dti tτ ω= −∞ +∞ ∫ g(t) g( )e d–i= −∞ +∞ ∫ 1 2π τ τωτ t = radiocarbon domain [BP] = [sec–1] t = calendar domain [sec] i = imaginary unit t = 2pn = angular frequency [sec–1] ˆ ˆ Bernhard Weninger, Kevan Edinborough 558 rescaling need not bother us. The rescaling of [BP]–> [sec–1] would imply rewriting the Libby equation, but in which neither we (as archaeologists) nor the 14C-labs have any practical interest. To this dimensional point, one of the present authors (BW) must admit some long-standing mistaken thinking. The notion that the 14C-scale is itself dimensionless with [BP]=unit [1] for the 14C-scale is erroneous. This only applies to the dimensional product of the 14C-scale and the calendar time scale. [8] What is the difference between scale change and domain switch? To preclude further misunderstanding we note that, in our terminology, the change of values between [BP] and D14C is a scale change, but this is a different concept than a domain switch. We use the term domain switch according to Hilbert space theory, where it denotes a change between orthogonal scales. Again, lan- guage is important. As noted above, the use of the word domain to denote a change in scale for functions that are not orthogonal (e.g. [BP], D14C, and F14C) in recent radiocarbon literature (Heaton et al. 2020) is definitely a correct use of such mathematical terms. It is nevertheless notable for leaving aside the possibil- ity that 14C-calibration can be described as a Fourier transform. [9] Fourier transform 14C-age calibration theorem It is important that the structure of the paired equations (Eq. 3 and Eq. 4) provides proof for the follow- ing theorem: following the initial Gaussian area normalization (Eq. 2), under the Fourier transform there is no need for further normalization of any of the 14C-distributions. The validity of this theorem follows im- mediately from the internal structure of Equation 3 and Equation 4. Although we have here two equations, both contain the same function. This function is either called g(t) (in both equations), or else called ĝ(t) (in both equations). This proof is surprisingly simple, even trivial. It is all the more important, however, in view of all the many studies (all the way back to the 1970’s, but still continuing today) that erroneously apply secondary normalization (or weighting) to calibrated 14C-ages. Also, in view of so many studies that have un- successfully attempted to correct the data for the slope-variability of calibration curve, it is perhaps useful that we formulate this theorem as a software programming rule: to avoid chronological distortion, additional normalization (beyond the initial norm=1 setting of the input Gaussians), is forbidden under the Fourier trans- form. It must be avoided (in both domains) under all circumstances. [10] Definition of 14C-dating probability under the Fourier transform Under the Fourier transform, it is finally possible to introduce a satisfactory (two-dimensional) concept of 14C-dating probability. The main underlying condition for this concept is that both domains contribute equal- ly to the probability measure. In mathematical terminology, the existence of this probability is guaranteed because there exists a unique 2D-area in the calibration system that has a scaling dimension [sec] on the ca- lendric scale, and scaling dimension of [sec–1] on the 14C-scale. When shown graphically, and in particular when projected as a plateau-rectangle onto the calibration curve, the 2D-area that can be derived – by dif- ferent methods – as product function ĝ(t)·g(t) has all set-theoretical properties (in particular: additivity), as well as dimension [sec–1]·[sec]=[1], that are needed to define a probability. This can be achieved by simulta- neous re-interpretation of the 14C-scale distribution and the corresponding (unnormalized) calibration distri- bution as wave functions. It is then possible to define what we call a gauge probability for calibrated 14C- ages (cf. main text). This is in accordance with the Born rule, which is the standard procedure in quantum theory, where it is used to define the (measurable) probability P(x,t) of finding a wave-particle in location x and at time t based on the squared amplitude of wave function y(x,t): Born rule: probability definition. [Eq. 5] P = Normalised probability defined for squared ampli- tude of wave-function y(x,t), with optional scale factor N to cover a finite number of wave-particles. N is hereby an initially unknown constant that is independent of x, but which can be determined by a sim- ple requirement: let us assume that the probability of finding the wave-particle somewhere is unity i.e. N=1. As indicated in Equation 6, under such conditions (when applied to Gaussian wave-functions) the Born rule cancels out the troublesome complex numbers eiwt and e–iwt contained in Equation 3 and Equation 4. This is because eiwt · e–iwt = 1. In analogy for a set of N 14C-dates, it is in consequence possible to assign a ‘Dating Probability’ based on the product of the two wave functions ĝ(t) and g(t): p(x,t) N (x,t) dt= = −∞ +∞ ∫ Ψ 2 1 Bayesian 14C-rationality, Heisenberg uncertainty, and Fourier transform> the beauty of radiocarbon calibration 559 Born rule: probability definition. [Eq. 6] P = Normalised probability defined for product of wave-function y(x,t) and its complex conjugate y*(x,t) under Fourier transform. As a reminder, ĝ(t) and g(t) represent the same date, but viewed from the two complementary domains t (calendric scale) and t (14C-scale). In the end, it is the pleasing properties of the Fourier transform on which the Born rule is based. [11] Definition of gauge probabilities for 14C-dates under the Fourier transform Functions that satisfy the conditions of Equation 6 are known as square-integrable. In quantum theory, this is one of the most important properties of the wave-function used to describe any wave-particle. As men- tioned above, this condition guarantees the existence of the particle, in the sense that it can be found at some time, and measured at some place, even though the particle has wave-properties. Similarly, if we now define ĝ(t) and g(t) as wave-functions, the condition of square-integrability allows us to introduce a genuine (i.e. properly normalized) probability, if only for the product of the two wave functions ĝ(t)·g(t): Born rule applied to radiocarbon dates: P = Normalised probability defined for the product [Eq. 7] of the Gaussian function ĝ(t) in the 14C-domain and its twin associated cal-domain function g(t) under Fourier transform. Note the symmetry of Equation 7 and Equation 6. In contrast, for the functions ĝ(t) and g(t) when viewed separately (on their respective domains), it is not possible to define a measurable probability. For the very clear formulation of this important property, Max Born was awarded the Nobel Prize in Physics (although rather late, in 1954). Yet, that does not mean that all related questions are today satisfactorily resolved. As for 14C-analysis, even under the Born rule, the dif- ferent concepts of probability (i.e. quantum theoretical just as Bayesian: both gauged probabilities and like- lihoods) still suffer under the same restriction. The problem is that the quadratic integrability in particular of the archaeological study function g(t), as requested in Equation 5, is often not ensured. When ensured, the task of this property is to guarantee that already (by itself) the calibrated wave function g(t) would re- present a genuine probability. Unfortunately, we cannot resolve the many multiple readings of the 14C-wave function on the wiggles of the calibration curve. The main problem to be resolved is that – using the per- haps better-known terminology of Bayesian 14C-calibration – we have the existence of two very differently scaled concepts of probability, but which are typically used in parallel. These two concepts are, (1), the 14C- measurement itself. For researchers in the lab, 14C-ages have a well-defined unit=1 probability, namely as measurement represented on the 14C-scale. However, (2) the very same Gaussian – from archaeological perspective – is seen as a wave-function that can be alternatively (or simultaneously) represented both on the 14C-domain as well as on the calendric domain. When understood as representing an archaeological date (and not alone a 14C-measurement) both wave-functions have ill-defined square-integrability. On both domains, this is of course mainly (but not only) due to the non-monotonous character of the calibration curve. It is particularly important to remember – one can easily overlook or even disbelieve this fact – that 14C-calibration is a Fourier transform even for a linear calibration curve. In conclusion, and with great admi- ration and due respect for all researchers involved in radiocarbon dating, the 14C-calibration does not have its foundation in Bayesian statistics but in mathematical physics. p(x,t) x,t x,t)dxdt = 1= ∫∫ ∗Ψ Ψ( ) ( p(t, ) g t g )dtd = 1τ τ τ= ∫∫ ( ) (ˆ