Approved For Release 2000/08/11 : CIA-RDP96-00792ROO0500450002-2 Volume 67A, number 3 PHYSICS LETTERS S-MATRIX, FEYNMAN ZIGZAG AND EINSTEIN CORRELATION 0. COSTA DE BEAUREGARD Instizut Henri Poincari, 75005 Paris, France Received 2 September 1977 Revised manuscript received 15 June 1978 7 August 1978 An inherent binding between Einstein correlations and the S-matrix formalism entails full relativistic covariance, corn- plete time symmetry, and spacelike connexions via Feynman zigzags. The relay is in the past for predictive correlations between future measurements, and in the future for retrodictive correlations between past preparations (Pflegor and Mandel). An analogy and a partial binding exist between intrinsic symmetry together with factlike asymmetry of (1) "blind statistical" prediction and retrodiction (retarded and advanced waves, information as cognizance and as will) and (2) posi- tive and negative frequencies (particles and antiparticles). As advanced waves are required for completeness of expansions "Aqtiphysics"obeying blind statistical retrod ction should sh-aw up in avorovriate contexts. "varavsYcholoev" beine sub- natea as one of tnem To the Einstein [ 1,2] paradox *1 proper (correla- tion of measurements upon distant systems that have interacted) corresponds a time-reversed Einstein para- dox (correlation of distant preparations that will inter- act), both very well sustantiated experimentally [3,4]. As implied in the mathematics, and as now demon wch facts have been dreaded. 15-7j a-n-d -are sttr~.teO still felt [81 as extremely paradoxical. To Einstein [51 they meant "telqp~,t y. , to _c. to. ing~~ [61 14 magic", to de Broglie [7] "ypsgtting our acceptCd yijew4 con- r ac and-time". Their existence heralds the advent of a new paradigm, that is, the wording and conceiving of a Weltanschauung strictly taylored after the mathematics. What is intended here is: (I ) A concise and "manifestly covariant "fornializa- tion of the mathematics. This has not yet been done, but it should be, because, although the paradox can be expressed in non relativistic quantum mechanics [21, it is in relativistic quantum mechanics that its full sig- nificance shows up. (2) 7he outlining of a Weltanschauung taylored strictly after the mathematical symmetries, the grand Paradox, "a suprising but perhaps true statement" (mean- ing no. I in all dictionaries). Copernicus' heliocentfism has been a paradox. example here being Einstein's interpretation of the group structure of the Lorentz-Poincard formulas. Pytting things bluntly, the monster awoken as early as 1927 by Einstein [I I is born from the union of in- trinsic mathematical time symmetry with Born's prin- ciple of adding partial amplitudes (rather than probabil- ities). And, as both genitors have a well established paradoxical" reputation, what of the offspring? 1. Concise and manifestly covariant formalization of the Einstein (predictive and retrodictive) correla- tions. First we need a general formalization of the n- tuple Einstein correlations *2. We assume the existence of a state vector expandable in the form cu.. loi) I 0j)... , where the 10)'s 10)'s, ..., span disjoint Hilbert spaces, and also of an operator M that is the direct product of hermitean operators m, p,..., operating respectively on the 10)'s, 42 Garuccio and Selleri 19] and Costa de Beauregard 191 have given the formula in the more restricted, diagonal form: 14)) 1 Eci 10i) 1*i). 171 Approved For Release 2000/08/11 : CIA-RDP96-00792ROM09450002-2 Approved For Release 2000/08/11 : CIA-RDP96-00792ROO0500450002-2 Volume 67A, number 3 The mean value of the magnitude M.- PHYSICS LETTERS (4) 1 M 14)) 0,...cij ... (or I m I oi)< ~i, lp I (2) contains a fully diagonal contribution having the form of a classical sum of partial probabilities, plus a com- posite, off diagonal, interference style, contribution, entailing the "paradoxical" Einstein correlation. By interpreting, with Dirac [101 and Land6 [I I any expansion 10 E c4 10 ) in the form (A 14)) = Ej I J i (A 000), we shall show that, in the Schwinger- Feynman interaction picutre, the transition amplitude ~411 I(D2) = (*I 1U(D1 (P2 Ul~D2), (3) between an "initial" 14fl) ~_ 42(ol )) and a "Final" '(D2) 14)(02)) state is of the form (I), where U de- notes that specification of the unitary evolution oper- ator leading from ul to 02 - Introducing a complete set of orthogonal projectors 10) (81 adapted to the problem considered (for ex- ample, predictive correlation polarizations [31, or tetro- dictive correlation occupation numbers [41 ) we re- write the amplitude (3) as I ):= E (*l I 0)(E)l'P2 (4) (D2 E) In a predictive problem we interpret [10, 11 ] the (81(b2Ys as the components of the final state and the ~*l 10)'s as the coefficients of the expansion. In a retrodictive problem we interpret the (191*1)'s as the components of the initial state and the ((P2 1 E)Ys as the coefficients of the expansioh. Both expressions are of the form (1). For example, in quantum electro- dynamics, the 10)'s in eq. (1) are the photon 1A), and the electron I ~), and the positron 141), states 1121. We may now interpret (TOP2) like (E) I (D2), regard- ing (4f, as a label like we interpret (0.1* One logically missing link in the Schwinger- Feynman formalism was an explicitly covariant defini- tion of the 10(o)) states used initially and finally, and of their hermitean scalar product, etc., by means of 3-fold o integrals. This has been given 113 1. Summarizing this section: (I) In relativistic quan- tum mechanics, the Einstein 111 correlations between "presently " separated systems are tied by Feynman zigzags. (2) The relay is in the past for predictive cor- relations between measurements, and in thefuturefor 172 Approved For Release 200'd/08/1 1/~ 7 August 1978 retrodictive correlations between preparations. (3) Full relativistic covariance and intrinsic time symmetry of these is thus madefortnally obvious - a point now needing a far from trivial epistemological discussion, as our ways of thinking are so macroscopically prejudiced! 2. Weltanschauung isomorphic to the -mathematics. The time asymmetry of the quantal measurement is of macroscopt .c orl.g7.12 *3 as it implies the idea Of a Tepeti- tion of the process, and, thus, a reference to the itatis- ticalfrequency approach to probability. In an individ- ual qu.-ntal event (such as the reception of one photon in the Pflegon-Mandel experiment) there is, and there can be, no intrinsic time asymmetry; but of course, in this case, what is needed is the Bayesian [ 151 approach to probability - the one consistently (although implic- itly) used in this paper. It is now well known [ 16] that the macroscopic time asymmetry (be it expressed as "blind retrodiction forbidden" [ 171, or "increasing probability" (Second Law), or "wave retardation") has a "factlike, not lawlike character" [ 181. Therefore, if by definition (macro)physics obeys the usual irrevers- ibility statements, and (macro)antiphysics the reversed statements, microphysics is just as neutral between physics and antiphysics as it is between particles and antiparticles. There is an analogy between the intrinsic symmeny versus factlike asymmeny of, on the one hand retarded and advanced waves, and of particles and antiparticles on the other hand. Not only is there an analogy, but also a partial binding, through the two expressions of the Jordan-Pauli propagator D(x - x) as (Dret - Dadv) and as (D, + DJ Not only is there this connection, but the very same argument (com- pleteness for an expansion), entailing the necessary presence of the D, and the D_ contributions in the Fourier expansion of solutions of the covariant wave equation, does entail that of the Dret and the Dad, contributions when solving the covariant position measurement problem [131. Therefore, by virtue of the very mathematics, it should be expected that, in appropriate contexts, antiphysical evolutions occasion- ally show up - very much like the positron or the anti- proton can be made to show up. Intrinsic time symmetry in x-space is analogous to intrinsic energy symmetry in k-space. The intrinsic time symmetry "paradox" is rooted *3 Davies 1141 makes the point quite clearly. CIA-RDP96-00792ROO0500450002-2 Approved For Release 2000/08/11 : CIA-RDP96-00792ROO0500450002-2 978 'Ietry low n, as liced! Volume 67A, number 3 deeper than in its Loschmidt and Zermelo versions: inside the probability calculus itself, because, even if the transition probabilities are symmetric between states (as in card shuffling or in radioactive decay) "blind statistical prediction" is physical while "blind statistical retrodiction" [161 would be antiphysical. Now, Aristotle's concept on the information I of cybernetics is twofold: gain in knowledge and orgam .z- ing po wer. Tile learning transition N - 12 occurs at, say, the reception of a message carrying a negentropy N and the organizing transition 11 - N at the emission. Not- withstanding the defacto inequalities 11 > N > 12 (Second Law) there is an intrinsic symmetry between the two transitions N # 1, and it is in a one-to-one connection with the intrinsic symmetry between en- tropy increasing (or physical) and entropy decreasing (or antiphysical) evolutions, and also between "blind statistical" prediction and retrodiction. Now, the "wavelike probability calculus" (initiated in 1926 by Born in quantum mechanics) brings in a one-to-one binding between retarded waves and blind statistical prediction on the one hand, advanced waves and blind statistical retrodiction on the other - a fact clearly emphasized by Fock [ 19]. A hermitean scalar product such as (q/1 14)2) is symmetric in 1*1 and *2), but it can be thought of, and used, asymmetrically, either as the projection of 1*1) upon 14Y, called col- lapse (for prediction via retarded waves with sources on 02) or as the projection of 14)2) upon 41), which can be called anticollapse (for retrodiction via advanc- ed waves with sinks on o1). "Irreversibility of quantal measurements" comes in via repetition, that is, with the frequency interpreta- tion of probability. It then belongs to (macro)physics, and it isfactlike, not lawlike. [18) Itcomesinviavon Neumann's ensembles and density matrix. In fact von Neumann derives entropy increase from wave retarda- tion (after the time t = 0 of the measurement), while of course entropy decrease would follow from wave advance (before t = 0) [20]. This is another wording of Fock's [ 191 statements *4. EAially , whpt would the pheqomenology,of ad- vanced waves, decreasing probability, blind statistical retrodiction, and info- ati ' ~M on as organic Un ppwer, *4 Factlike time asymmetry in the S-matrix formalism is ob- tained via the integration contour in k-space, by definition of the Feynman propagators for virtual particles. PHYSICS LETTERS 7 August 1978 l2Qk.1jke? Exactly to wh t paraps chologists call pre- cognition and/or psychokinesis. Logically these phe nomena should show up, no less than thermodynamical yp~g~essingfl!~ctlu,at7ion_s - Consciousness has two faces symmetric to each other: cognizance and will. hoth sho'uld- show up it? the quantal measurement process, ties. is of )eti- atis- vid- _)ton here in oach plic- that is 9 has if ers- .,rsed ,id msic ere two XI) ive sion- anti- to References [11 A. Einstein, in: Rapports et discussions du Se Conseil Solvay (Gauthier Villars. Paris, 1928) p-253. [21 A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 4 7 (1935) 777. 131 S.J. Freedman and J.F. Clauser, Phys. Rev. Lett. 28 (1972) 938; J.F. Clauser, Phys. Rev. Lett. 36 (1976) 1223: E.S. Fry and R.L. Thompson, Phys. Rev. Lett. 37 (1976) 465; M. Lamehi-Rachti and W. Mittig, Phys. Rev. DI 4 (1976) 2543; A.R. Wilson, J. Low and D.K. Butt, J.Phys. G2 (1976) 613. 14J R.L. Pflegor and L. Mandel, Phys. Rev. 159 (1967) 1084 (see formula 18 and discussion); J. Opt. Soc. Am. 58 (1968)946. 151 A. Einstein, in: Einstein, philosopher, scientist, ed. P.A. Schilpp (The Library of Living Philosophers, Evanston, IL) pp.85, 683. 161 E. Schr6dinger, Naturwiss. 23 (1935) 844; see p.845. 17] L. de Broglie, Une tentative d'interpr6tation causale.... de [a m6canique ondulatoire (Gauthier Villars, Paris, 1956) p.73. 181 See for example: B. d'Espagnat, Conceptual foundations of quantum mechanics, 2nd ed. (Benjamin, Reading, MA, 1976); A. Shimony, Epistemological letters (Lausanne, 1976) pp. 1 -5. (91 A. Garuccio and F. Selleri, Nuovo Cimento 36B (1976) 176; 0. Costa cle Beauregard, Lett. Nuovo Cimenio 19 (1977.) 113. , [101 P.A.M. Dirac, The principles of quantum mechanics, 3rd ed. (Oxford Clarendon Press, 1948) p.79. (III A. Lam:16, New foundations of quantum mechanics (Cambridge Univ. Press, 1965) P-83. 1121 For an example see: 0. Costa de Beauregard, Phys. Lett. 60A (1977) 93. 1131 0. Costa de Beauregard, Pr6cis de m6canique quantique relativiste (Dunod, Paris, 1967); Nuovo Cimento 42B (1977)41. 114 1 P.C.W. Davies, The physics of time asymmetry (Surrey Univ. Press, 1974) pp. 174-175. 173 Approved For Release 2000/08/11 : CIA-RDP96-00792ROO0500450002-2 Approved For Release 2000/08/11 : CIA-RDP96-00792ROO0500450002-2 Volume 67A, number 3 (151 See in this respect: R.T. Cox, The algebra of probable inference (The John Hopkins Press, Baltimore, 1961). esp. p.2; or M. Tribus, Rational descriptions, decisions an dc- signs (Pergamon, 1969), esp. pp.65-66. 1161 For an overall view and a guide to the literature, see: 0. Costa de Beauregard, Proc. Intern. Cong. for logic, methodology and the philosophy of science, ed. Y. Bar PHYSICS LET-FERS 7 August 1978 Hillel (North-Holland, Amsterdam, 1964), or Studium Generale 24 (1971) 10. 117 ] S. Watanabe, Rev. Mod. Phys. 27 (1952) 26, 40, 179. (181 H. Mehlberg, Current issues in the philosophy of science, eds. Feig] and Maxwell (Holt, Rinehart, Winston, New York, 1961 ) p. 105. (191 V. Fock, DokI. Akad. Nauk. SSSR 60 (1948) 1157. (201 0. Costa de Beauregard, Cah. de Phys. 12 (1958) 317. 174 Approved For Release. 2000/08/11 : CIA-RDP96-00792ROO0500450002-2