Download Transition amplitudes versus transition probabilities and a

"Quantum Concepts
Eds.R.Penrose
Clarendon
in Space and Time'
and C.J.lsham
e2Jrivt
15.1. Introduction
The rise of quantum physics, very surprisingly, has exerted little influence
on our views about space and time. With the exception of quantum
gravity, all theories assume that quantum processes occur in the spacetime of classical physics, and even in quantum gravity the modifications
are presumably restricted to regions of space-time of the size of the
Planck length.
In this chapter I argue that the very structure of all quantum theories
suggests a revision of the classical notions of space and time. I will
present evidence that two copies of space-time, rather than one, are the
proper arena for all quantum processes. At the heart of this observation
lies the very well known fact that every set of equations and formulae in
quantum theory, from which all the transition amplitudes are determined, may always be written in two equivalent forms, differing by
complex conjugation. We obtain one set from the other by reversing the
sign of the imaginary unit i. This is clearly seen already in the simplest
case of the non-relativistic quantum mechanics. The sign of i may be
chosen according to the following two conventions:
ili at'll = H'II,
-ili at'll = H'II,
[x, pI
= iii,
= [F, Hl,
[x, p] = -iii,
-iliP = [F, HI.
eiEtlh.
P=A* ·A,
Iwo Bialynicki-Birula
II. Opposite convention
or
(15.2)
When the two formulations were compared, it became obvious that one
of the signs had to be reversed to achieve consistency. Of course, both
formulations, I and II, are physically completely equivalent; they give
identical transition probabilities. The transition probabilities are defined
as squared moduli of transition amplitudes,
Transition amplitudes versus
transition probabilities and a
reduplication of space-time
I. Standard convention
227
occur, since in both formulations of quantum mechanics, considered
independently from each other, it is natural to describe the time evolution through the harmonic factors
Press, Oxford 1986
15
iltP
Transition amplitudes versus transition probabilities
(15.1)
Convention I originated with matrix mechanics (Born and Jordan 1925),
while convention II was used originally by Schr6dinger (1926, 1927) in his
ftrat papen on wave mechanics. The clash of the signs was bound to
(15.3)
so that a change of convention results merely in an interchange of A and
A*.
The interchange of the two formulations can also be effected by the
sign reversal of the Planck constant Ii. This is the first indication that the
existence of the two formulations may have something to do with
space-time properties of quantum mechanics. After all, the Planck
constant is a dimensional quantity that depends on the units of length and
time. In particular, it changes its sign when the unit of time is reversed.
As a result of this, the interchange of the formulations I and II is
essentially equivalent to the Wigner time reversal (Bialynicki-Birula
1957).
The association of the transition amplitude A with the process evolving
forward in time and of the complex conjugate amplitude A * with the
time-reversed process enables one to view the transition probability P as
describing a process that evolves around a loop in time. Such loops in
time were introduced long ago (Schwinger 1960, 1961) as a mathematical
device that helps to evaluate expectation values. I believe that these
loops in time should be taken more realistically and I shall present here
some tentative ideas on this subject.
15.2. Probabilities and two-sided diagrams
Let me begin with non-relativistic quantum mechanics. The transition
probability for a particle to travel from Xo, to to x, t can be expressed as a
product of two path integrals (Feynman and Hibbs 1965):
J
p(xt; Xoto) = DXl
f
DX2
exp~ (Sl - Sz)·
(15.4)
This formula is explicitly symmetric under complex conjugation and
under the change of the sign of It, As a result, expression (15.4) is
Quantum concepts in space and time
228
Transition amplitudes versus transition probabilities
II
229
the new variables, the formula (15.4) reads
xl!
p(xt; Xo4J)
J J
= Dx Dx exp[ -i fdt(mX + VV(x» . x + 0(;"2) ]
to
J
= Dx <5(tiix + VV) + 0(;"2),
(15.5)
where
Xl
xoT!o
Fig. 15.1
invariant under the interchange of the two conventions. However, if
expression (15.4) is to be considered as more fundamental than the single
path integral representing the transition amplitude, then both trajectories
Xl and Xz must be treated on an equal footing. This can be achieved in a
natural way if we introduce two copies of space-time and we sum over all
trajectories lying in both copies. At the end points both copies are
identified (Fig. 15.1). This identification is presumably related to the
classical nature of the measuring devices that determine the initial and
final co-ordinates of the particle. I shall say more about it later. The
perturbation expansion of the transition probability p(xt; Xo4J) further
explains my rationale for the reduplication of space-time in quantum
mechanics. Upon expanding the transition probability into a power series
in the potential V, we can represent each term by a simple two-sided
diagram (Fig. 15.2). The total probability is the sum of contributions
from all possible combinations of elementary scattering processes occurring in both copies of space-time.
In order to clarify a little the role of the classical description in the
context of the reduplicated space-time, I will perform a change of
variables from Xl and Xz to X and x (Bialynicki-Birula 1971). In terms of
+
+
+
PlI·15.2
+
+
+ ...
= X + hX12,
Xz
=X
-
hX 12.
(15.6)
Formulae (15.5) and (15.6) show clearly that in the classical limit, when
;,,~ 0, the two copies of space-time collapse into one, and only one
trajectory, namely the one that is determined by the Newton equations,
contributes to the transition probability.
The same concept of a reduplicated space-time can be used also in
relativistic theories. As a matter of fact, the description based directly on
transition probabilities that makes use of two-sided diagrams of the type
shown in Fig. 15.2 has several advantages in quantum electrodynamics
and in other gauge theories. This description was introduced in the past
(Bialynicki-Birula 1970; Bialynicki-Birula and Bialynicka-Birula 1975) to
simplify the study of gauge invariance and renormalization. The twosided diagrams representing the lowest order expressions for the transition probability in the electron-electron (Meller), electron-positron
(Bhabha), and electron-photon (Compton) scattering are shown in Fig.
15.3. Two-sided diagrams are similar to the vacuum polarization
diagrams-they never have open lines. It is this very feature of the
two-sided diagrams that leads to significant simplifications, when the
corresponding expressions for the transition probabilities are written
down. In particular, it has been possible to eliminate all wave function
renormalization constants, which are a source of serious complications in
gauge theories, due to their gauge dependence. It is also known (Bloch
and Nordsieck 1937; Jauch and Rohrlich 1954; Lee and Nauenberg 1964)
that the transition probabilities, but not the transition amplitudes, are
free from infra-red divergences in theories with massless particles.
I have wondered for some time whether through all these cancellations
and simplifications nature is not trying to tell us something quite
important. Perhaps the two-sided diagrams representing, as they are,
directly observable quantities, are closer to reality. If so, we ought to
depart from the conventional approach to quantum scattering and treat
all processes as really going around a loop in time in the manner
represented by two-sided diagrams. The proper arena for quantum
processes would then be not one copy of Minkowski space-time, but
rather two copies of it-one copy for each side of the diagram. Once two
copies of space-time are introduced, we must face some burning
questions: how are these two copies joined together; what is the nature of
230
Quantum concepts in space and time
M011erscattering
Bhabha scattering
Transition amplitudes versus transition probabilities
renormalization procedure is adapted (mass-shell renormalization) to the
S-matrix, i.e. to infinite time-lapse calculations. The limits of the propagators, when t ~ ±oo, yield the same transition amplitudes as those
obtained more conventionally by Fourier transforming the truncated
propagators and taking the mass-shell values. This equivalence enables
one to interpret the two-sided diagrams as representing processes occurring in space-time. Originally (Bialynicki-Birula 1970) these diagrams
were introduced to describe transition probabilities in momentum space,
as they appear in the S-matrix calculations when the states are labelled
with the momentum vectors of the initial and final particles. Since the
space-time interpretation of the S-matrix plays an essential role in my
analysis, I shall describe in some detail the space-time description and
the momentum description of relativistic scattering processes.
The truncated propagators, whose Fourier transforms determine the
transition amplitudes in relativistic field theory, are related to the full
propagators through the relations.
G(Xb ...
Compton scattering
Fig. 15.3
this join; does this description apply equally well to particles? I must
admit that the discussions that I had in Oxford during the conference
have shattered some of my naive hopes for a simple solution to these
problems. However, I believe that the main idea has not been proven to
be worthless.
15.3. Transition probabilities on scri
In the full relativistic theory, in contradistinction to quantum mechanics,
the two copies of space-time should be joined together along the future
and the past infinity. This conclusion can be reached from the analysis of
the most promising relativistic theories-gauge
field theories. Transition
probabilities for finite times in such theories are not gauge invariant and
hence devoid of physical meaning. This is because the propagators of
particles carrying charges, Abelian or non-Abelian, are gauge dependent;
only their mass-shell values are gauge invariant. In addition, the whole
231
,xn)
= (-i)n
f
d4YI ... d4ynG(xb YI)· .. G(xn' Yn)F(Yb ... ,Yn),
(15.7)
where I have taken, for simplicity, a theory of only one type of scalar
particles. Instead of taking the Fourier transform of F and evaluating it
on the mass-shell, one may take the limits of the full propagator G, when
its arguments Xi are removed to infinity along the world lines of particles
moving freely with velocities VI. This follows from the fact that the
(renormalized) one-particle propagator G(x, y) behaves for large times
as the free Feynman propagator ~F and from the asymptotic form of the
free propagator:
J d''y ~F(Vt -
r', t - tl)F(r', t'; ...
- (Ym/2)( -2ni
It I \1'1-
)t->±oo ~
VZ)-3/Ze-im
2
ItI vr=v
f
d4ye±iP"YF(y,
... ),
(15.8)
where p denotes the four-momentum of the particle,
(p 1') = m(1 - vZ)-lIZ(1, v),
(15.9)
and r' and t' are the space and time components of the four-vector y.
For massless particles, the situation is quite different. Due to the
absence of dispersion in the propagation of the waves describing these
particles, the asymptotic behaviour of the massless propagators is
characterized by the first power of It I in the denominator. This factor is
common in the ma••ive and in the massless cases. In the massive case, we
232
Transition amplitudes versus transition probabilities
Quantum concepts in space and time
233
obtain an additional factor of Itl1/2 in the denominator due to the
spreading of the wave packet in the direction of the propagation. The
counterpart of formula (15.8) in the massless case has the form:
f
d'y D F(nt - r', t + to - tl)F(r' , t'; ...
)t-+±oo -
I)
( -8n2i It -1
x LoodWe=Fiwtfd4ye±ik'YF(y,
... )
(15.10)
where k is the four-momentum vector of the massless particle,
(klL) = w(l, n).
(15.11)
Thus, the full information about the scattering of massless particles is
contained in the limits of the propagators, when t ~ ±oo, if they are
known for all the values of the initial time to. In other words, one must
take the limit, when the advanced time v = t + r tends to +00, or the
retarded time u = t - r tends to -00 while the retarded time (in the first
case), or the advanced time (in the second case) is held fixed. Upon
comparing the formulae (15.8) and (15.10), we notice one additional
important difference: the limit of the propagator in the massless case does
not give directly the scattering amplitude of a particle with a given
four-momentum, but its Fourier transform with respect to energy. Still,
the limits (15.8) and (15.10) of the propagators with respect to all their
arguments determine all the elements of the S-matrix. Of course, prior to
taking the limits t ~ ±oo we must remove the kinematical factors in the
expressions (15.8) and (15.10): the appropriate powers of It! and the
oscillating factor describing the change of the phase of the wave function
along its free trajectory.
Our analysis has shown that the transition amplitudes in relativistic
quantum field theory are determined by the appropriately scaled values
of the propagators on the boundary of the Minkowski space-time.
Therefore, it seems natural to use in their analysis the concepts
introduced by Penrose (1964, 1968) in his 'conformal treatment of
infinity'. Such a conformal treatment, however, is not suitable to the
description of massive particles (d. also Binegar et al, (1982) for a
somewhat different approach). The description of those particles is
clearly pathological; their trajectories all begin at one point 1- (past
time-like infinity) and they all end at one point 1+ (future time-like
infinity). As a result, the limits of the (scaled) propagators at the points
1- and 1+ are ill-defined: they depend on the path along which these
points are approached. I shall restrict myself in my analysis to the
massless case and make only some speculative remarks about the massive
calo at tho ond.
Fig. 15.4
Transition amplitudes for massless particles are determined by the
(scaled) values of the propagators with their arguments lying in the future
null infinity .1> + and the past null infinity .1> -. As I have argued above, in
order to accommodate two-sided diagrams we need two copies of the
Minkowski space-time joined together along their common boundary, in
such a way that closed loops in time are possible. In the conformal
framework of Penrose there is essentially only one way of doing it: the
.1> + of the first copy is to be identified with the .1> - of the second copy and
vice versa (Fig. 15.4).
The two copies of Minkowski space-time with their boundaries
identified in such a manner form a hypertorus. The hypertorus is divided
into two halves by a distinguished light cone made of .1> + and .1> - -the
scri.
The basic tenets of relativistic quantum theory of massless particles
may be stated as follows. Quantum processes take place on the whole
hypertorus. The corresponding transition probabilities pick up contributions from trajectories lying in both halves of the hypertorus. The only
difference between the contributions from the first and from the second
half is that the imaginary unit i is replaced by -i. All transition
probabilities 'live' on the scri, where the two copies of the space-time
meet each other. These .statements follow from the standard formulation,
so that the physical content of the theory remains unchanged. However,
the transition probabilities appear here, in a natural manner, as functions
of the retarded and advanced times that label the points on the scri,
rather than as functions of the energies of the incoming and outgoing
particles. Otherwise, 'everything is just translated into an alternative, but
equivalent, mathematical framework. But the new framework suggests
different directions in which to proceed, from the ones which might seem
natural in a more conventional theory'. These words were written
(Penrose 1975) in connection with the twistor program, but I believe that
they are also applicable here. Additional roalonl for quotina Ponrole
Quantum concepts in space and time
234
stem from my expectations that the twistor formalism should prove very
useful in the analysis of the transition probabilities on the scri. After all,
twistors are a natural tool when the conformal properties playa role, and
one is led, nolens volens, to the conformal compactification in order to
describe two copies of the Minkowski space-time in 'a compact way'.
At the end I should say something about the mass, which is a source of
some embarrassment in every conformal theory. The standard treatment
of mass does not look very promising. In the conformally compactified
space-time all trajectories of massive particles converge at one point,
which in turn leads to a pathological behaviour of the transition
probabilities treated as functions on the scri. These functions must be
highly singular at 1+ and 1- in order to give different values at these
points for all different directions along which these points are
approached.
A possible way out of these difficulties is to assume that at the very
fundamental level massive particles always move with the speed of light.
The result of the mass term is a series of random 'kicks', which cause
discontinuous changes in the direction of the particle velocity
(Zitterbewegung). That the Dirac equation admits such an interpretation
is shown by Penrose and Rindler (1984) (zigs and zags) and by others in
connection with the Feynman path-integral approach to the Dirac
equation. In the most recent publication on this subject (Gaveau et al.
1984) the authors write: 'An electron propagates at the speed of light and
with a certain chirality (like a two-component neutrino), except that at
random times it flips both direction of propagation (by 180°) and
handedness: the rate of such flips is precisely the mass, m (in units
Ii::::: c :::::1)'. If this analogy with Brownian motion is assumed to be valid
also for interacting particles, then we could imagine that a random kick,
combined with a kick caused by the interaction, might send the particle
across the scri to the other copy of space-time. It is also possible that
quantum tunnelling plays a role here.
More hard facts, however, will have to be ascertained before we would
know whether these speculations can be taken seriously.
Acknowledgements
I would like to thank Z. Bialynicka-Birula, A. Bialynicki-Birula, C. Fronsdal, J. Kijowski,
A. Trautman and S. Woronowicz for many stimulating conservations. Discussions during
the Oxford Meeting, most notably those with R. Penrose, have contributed significantly to
the evolution of my ideas.
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-
(1970). Ph". Rlv. D 2, 2877.
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(1971). Ann. Phys. N.Y. 67,252.
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