# PHYSICS LETTERS - S-MATRIX, FEYNMAN ZIGZAG AND EINSTEIN CORRELATION

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S-MATRIX, FEYNMAN ZIGZAG AND EINSTEIN CORRELATION
0. COSTA DE BEAUREGARD
Instirut Henri Poincare, 75005 Paris, France
Received 2 September 1977
Revised manuscript received 15 June 1978
An inherent binding between Einstein correlations and the S-matrix formalism entails full relativistic covariance, com-
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
"antiphysics" obeying blind statistical retrodiction should show up in appropriate contexts, "parapsychology" being sub-
mitted as one of them.
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-
strated,. uch facts have been dreaded. 15- an are
still felt [8] as extremely paradoxical. To Einstein [5]
they meant "telepat y , to _c, to. tnger [61 "magic",
to de Broglie [7] "upsgtting our accepted yiew4 con-
cerning~space 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:
(1) A concise and "manifestly covariant "formaliza-
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 [2]
it is in relativistic quantum mechanics that its full sig-
nificance shows up.
(2) The outlining of a Weltanschauung taylored
strictly after the mathematical symmetries, the grand
*1 Paradox, "a suprising but perhaps true statement" (mean-
ing no. 1 in all dictionaries). Copernicus' heliocentrism has
been a paradox.
example here being Einstein's interpretation of the
group structure of the Lorentz-Poincare formulas.
Patting things bluntly, the monster awoken as early
as 1927 by Einstein [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?
]. 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
w _ Zi cis.. Itbi) I ty~ )... , (1)
where the 10's IiJ/)'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 IO)'s,
$2 Garuccio and Selleri 19] and Costa de Beauregard 191 have
given the formula in the more restricted, diagonal form:
Im) = Eciloi)hyi).
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The mean value of the magnitude M.?
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 [10] and Lande [ 1 1
any expansion 10 ) = Eic4 I0) in the form (A I4)) = Ei
(A 000), we shall show that, in the Schwinger-
Feynman interaction picutre, the transition amplitude
retrodictive correlations between preparations. (3)
Full relativistic covariance and intrinsic time symmetry
(4) IMIl) = 0,???c'i?.?(or Irn Ioi)< /i,lpI,j)... , of these is thus made formally obvious - a point now
(2) needing a far from trivial epistemological discussion, as
(W I4)2> _ (FYI IU(D1) = (`I'2UkP2),
(3)
our ways of thinking are so macroscopically prejudiced!
2. Weltanschauung isomorphic to the -mathematics.
The time asymmetry of the quantal measurement is of
macroscopic origin *3 as it implies the idea of a repeti-
tion of the process, and, thus, a reference to the statis-
tical frequency 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
between an "initial" I4,1> = I'2(ol )> and a "final" to probability - the one consistently (although implic-
14)2) 14)(02)) state is of the form (1), where U de- itly) used in this paper. It is now well known [ 16] that
notes that specification of the unitary evolution oper- the macroscopic time asymmetry (be it expressed as
ator leading from of to a2. "blind retrodiction forbidden" [ 171, or "increasing
Introducing a complete set of orthogonal projectors probability" (Second Law), or "wave retardation") has
IO) (01 adapted to the problem considered (for ex- a "factlike, not lawlike character" [ 181. Therefore, if
ample, predictive correlation polarizations [31 or retro- by definition (macro)physics obeys the usual irrevers-
dictive correlation occupation numbers [4]) we re- ibility statements, and (macro)antiphysics the reversed
write the amplitude (3) as statements, microphysics is just as neutral between
(4)
(WlI(D2 )'_ T (`I's as the
components of the initial state and the (4)218>'s as
the coefficients of the expansion. Both expressions
are of the form (1). For example, in quantum electro-
dynamics, the hp)'s in eq. (1) are the photon IA), and
the electron I~>, and the positron 141), states 1121.
We may now interpret (W114)2) like (0I(D2), regard-
ing (Wi 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 I?(o)) states used initially and finally, and
of their hermitean scalar product, etc., by means of
3-fold o integrals. This has been given [13 1.
Summarizing this section: (1) 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 the future for
physics and antiphysics as it is between particles and
antiparticles. There is an analogy between the intrinsic
symmetry versus factlike asymmetry 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+ + D_ ). Not only is there
this connection, but the very same argument (com-
pleteness for an expansion), entailing the necessary
presence of the D.1. and the D_ contributions in the
Fourier expansion of solutions of the covariant wave
equation, does entail that of the Dret and the Dadv
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.
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terry
ow
-ft, as
.iiced!
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 organiz-
ing power.
The 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 de facto inequalities I1 > 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 (114)2) is symmetric in 1*1 and 14)2),
but it can be thought of, and used, asymmetrically,
either as the projection of 1'1) upon 14)2), called col-
lapse (for prediction via retarded waves with sources
on 02) or as the projection of 14)2) upon IW1), 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) It comes in via von
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.
Finally, whet would the phenomenology of ad-
vanced waves, decreasing probability, blind statistical
retrodiction, and information as organiztn power,
*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.
12ok.1ike? Exactly to what parapsychologists call pre-
cognition and/or psychokinesis. Logically these phe_
nomena should show up, no less than thermodynamical
progressing fluctuations - which indeed they are.
Consciousness has two faces symmetric to each
other: cognizance and will. Both should show up in the
quantal measurement process.
tics.
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(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. 47
(1935) 777.
[31 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. D14 (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.
[51 A. Einstein, in: Einstein, philosopher, scientist, ed. P.A.
Schilpp (The Library of Living Philosophers, Evanston,
IL) pp.85, 683.
161 E. Schrodinger, Naturwiss. 23 (1935) 844; see p.845.
17] L. de Broglie, Une tentative d'interpretation causale....
de Ia mccanique ondulatoire (Gauthier Villars, Paris,
1956) p.73.
(8) 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.
(9) A. Garuccio and F. Selleri, Nuovo Cimento 36B (1976)
176;
0. Costa de Beauregard, Lett. Nuovo Cimento 19 (1977.)
113. ,
[101 P.A.M. Dirac, The principles of quantum mechanics, 3rd
ed. (Oxford Clarendon Press, 1948) p.79.
[Ill A. Lando, New foundations of quantum mechanics
(Cambridge Univ. Press, 1965) p.83.
(121 For an example see: 0. Costa de Beauregard, Phys. Lett.
60A (1977) 93.
1131 0. Costa de Beauregard, Precis de mccanique quantique
relativiste (Dunod, Paris, 1967); Nuovo Cimento 42B
(1977) 41.
(14 1 P.C.W. Davies, The physics of time asymmetry (Surrey
Univ. Press, 1974) pp. 174-175.
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PHYSICS LETTERS 7 August 1978
(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 de-
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
Hillel (North-Holland, Amsterdam, 1964), or Studium
Generale 24 (1971) 10.
(171 S. Watanabe, Rev. Mod. Phys. 27 (1952) 26, 40, 179.
(181 H. Mehlberg, Current issues in the philosophy of science,
eds. Feigl and Maxwell (Holt, Rinehart, Winston, New
York, 1961) p. 105.
(191 V. Fock, Dokl. Akad. Nauk. SSSR 60 (1948) 1157.
120] 0. Costa de Beauregard, Cah. de Phys. 12 (1958) 317.
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