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Volume 67A, number 3
PHYSICS LETTERS
7 August 1978
S-MATRIX, FEYNMAN ZIGZAG AND EINSTEIN CORRELATION
O. COSTA DE BEAUREGARD
Institut Henri Poincaré, 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, such facts have been dreaded. [5-7] and are
still felt [8] as extremely paradoxical. To Einstein [5]
they meant "telepathy", to Schrödinger [6] "magic",
to de Broglie [7] "upsetting our accepted views 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-Poincaré formulas.
Putting things bluntly, the monster awoken as early
as 1927 by Einstein [1] 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
($) = [ci. ... 10;14;).
(1)
where the lø)'s lu)'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 lø)'s, Iv)'s, ....
+2 Garuccio and Selleri [9] and Costa de Beauregard [9] have
given the formula in the more restricted, diagonal form:
ΙΦ) = Σε; ορίψη.
171
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PHYSICS LETTERS
7 August 1978
The mean value of the magnitude M:
(ΦΙΜΙΦ)=
(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 [10] and Lande [11],
any expansion ($4> = Σ,c Ιφ;) in the form (Alφ) = Σ;
(AljXjlø), we shall show that, in the Schwinger-
Feynman interaction picutre, the transition amplitude
(3)
between an "initial" \\1>=/2(01)> and a "final"
12> = l+((2)> state is of the form (1), where U de-
notes that specification of the unitary evolution oper-
ator leading from 1 to 02.
Introducing a complete set of orthogonal projectors
IO> <Ol adapted to the problem considered (for ex-
ample, predictive correlation polarizations [3] or retro-
dictive correlation occupation numbers [4]) we re-
write the amplitude (3) as
(+1+2)=Cky1 10xe(¢2).
(4)
In a predictive problem we interpret [10,11] the
(Ol+2}'s as the components of the final state and the
(1 10)'s as the coefficients of the expansion. In a
retrodictive problem we interpret the (Oly,)'s as the
components of the initial state and the (+2l℮)'s as
the coefficients of the expansion. Both expressions
are of the form (1). For example, in quantum electro-
dynamics, the lø)'s in eq. (1) are the photon IA), and
the electron IT>, and the positron lw), states [12] .
We may now interpret (\11+2> like (@1+2), regard-
ing (1 as a label like we interpret (O."
One logically missing link in the Schwinger-
Feynman formalism was an explicitly covariant defini-
tion of the lø(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].
Summarizing this section: (1) In relativistic quan-
tum mechanics, the Einstein [1] 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
retrodictive correlations between preparations. (3)
Full relativistic covariance and intrinsic time symmetry
-
of these is thus made formally obvious - a point now
needing a far from trivial epistemological discussion, as
our ways of thinking are so macroscopically prejudiced!
·
1.
j
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 quantal 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 [15] 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" [17], or "increasing
probability" (Second Law), or "wave retardation") has
a "factlike, not lawlike character" [18]. 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
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+ 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 [13]. 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 [14] makes the point quite clearly.
172
* This sentence is thereforees not mive , and I dont like it
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Volume 67A, number 3
PHYSICS LETTERS
7 August 1978
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" [16] would be antiphysical.
Now, Aristotle's concept on the information / 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 /1 - N at the emission. Not.
withstanding the de facto inequalities /1 > 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 (1+2) is symmetric in N/1 and ld2),
but it can be thought of, and used, asymmetrically,
either as the projection of N+1> upon l+2), called col-
lapse (for prediction via retarded waves with sources
on 02) or as the projection of l$2) upon !¥1), 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 is factlike, 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 [19] statements #4.
Finally, what would the phenomenology of ad-
vanced waves, decreasing probability, blind statistical
retrodiction, and information as organizing 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.
look like? 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.
References
[1] A. Einstein, in: Rapports et discussions du Se Conseil
Solvay (Gauthier Villars. Paris, 1928) p.253.
[2] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47
(1935) 777.
[3] 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.
[4] R.L. Pflegor and L. Mandel, Phys. Rev. 159 (1967) 1084
(see formula 18 and discussion); J. Opt. Soc. Am. 58
(1968) 946.
[5] A. Einstein, in: Einstein, philosopher, scientist, ed. P.A.
Schilpp (The Library of Living Philosophers, Evanston,
IL) pp.85, 683.
[6] E. Schrödinger, Naturwiss. 23 (1935) 844; see p.845.
[7] L. de Broglie, Une tentative d'interprétation causale ....
de la mécanique 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;
O. Costa de Beauregard, Lett. Nuovo Cimento 19 (1977)
113.
[10] P.A.M. Dirac, The principles of quantum mechanics, 3rd
ed. (Oxford Clarendon Press, 1948) p.79.
[11] A. Landé, New foundations of quantum mechanics
(Cambridge Univ. Press, 1965) p.83.
[12] For an example see: O. Costa de Beauregard, Phys. Lett.
60A (1977) 93.
[13] O. Costa de Beauregard, Précis de mécanique quantique
relativiste (Dunod, Paris, 1967); Nuovo Cimento 42B
(1977) 41.
[14] P.C.W. Davies, The physics of time asymmetry (Surrey
Univ. Press, 1974) pp. 174-175.
173
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Volume 67A, number 3
:
PHYSICS LETTERS
7 August 1978
[15] 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.
[16] For an overall view and a guide to the literature, see: O.
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.
[17] S. Watanabe, Rev. Mod. Phys. 27 (1952) 26, 40, 179.
[18] H. Mehlberg, Current issues in the philosophy of science,
eds. Feigl and Maxwell (Holt, Rinehart, Winston, New
York, 1961) p. 105.
[19] V. Fock, Dokl. Akad. Nauk. SSSR 60 (1948) 1157.
[20] O. Costa de Beauregard, Cah. de Phys. 12 (1958) 317.
174
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