"States and observables in relativistic quantum field theories"

appeared in Phys. Rev. D, 15 p,3316, June 1980

 by Yakir Aharonov and David Albert.

Notes by Jack Sarfatti.

Aharonov and Albert show that Bohr's meta theory of the quantum (i.e., the Copenhagen interpretation) has a serious conflict with Einstein's special theory of relativity.


in contrast to the nonrelativistic case, it is not possible to define the quantum state of a system in relativistic quantum field theories, because in this latter case no consistent description of how the state changes as the result of a measurement can be developed.
We see that the problem centers on the notion of the nonunitary "collapse" of the state in the measuring process.
no relativistically satisfactory version of the collapse postulate can be found.
In contrast, there is no collapse in Bohm's theory of the quantum, although Bohm's theory also has an analogous problem with special relativity for individual events though not for their statistical averages.
the capacity of the theory to predict probabilities is to some extent independent of its capacity to define a state for a given system. Indeed the main result of the present work is that although relativistic field theories have the former capacity, they lack the latter one.
The authors also show that "various nonlocal properties of certain systems" can be "directly measured."


Those state histories which may be checked by experiment will not transform correctly between different frames and, conversely, those which are defined so as to transform correctly will lack the capacity to be verified by experiments.

P(a,b,c,...;T1,T2,T3/g,h,i, ...;t1,t2,t3)

is the conditional transition probability that if the results of measurements of observables A,B,C, are a,b,c, ... at times T1,T2,T3, ... respectively, that the results of measuring G,H,I at times t1,t2,t3, will be g,h,i respectively.


 (i) The theory "contains a covariant prescription" for calculating transition probabilities.

 (ii) It is possible "everywhere in the future" ... to define a unique succession of states ... a state history ... such that psi evolves in t in accordance with covariant dynamical equations of motion at all times except when the system is being measured.

 (iii) Psi transforms in accordance with the requirement that the equations of motion be covariant.

 (iv) If C psi(t) = c psi(t),


P(a,b;T1,T2/c:t) = 1



Note that the operator C which satisfies this relation will depend both on the time and frame of reference.
All of these desirata are obeyed for the Galilean group of nonrelativistic quantum mechanics.

 The measurement in (iv) is "nondemolition", it

will not disturb the history of the state in any way"... Indeed a limit can be approached in which the state is checked at every instant by a nondemolition experiment, and this we will call a monitoring of the state.
The "atomic" conditional transition probability that all others can be built from is


P(q:T/b:t) = |psi(a;T/b;t)|^2 (1)

The equation of motion in the Galilean case is the Schrodinger equation. The additional boundary condition is


A psi(a;T/b:t=T) = a psi(a;T/b:t=T) (2)

A very important case is when a measurement of C is made after a measurement of A but before a measurement of B then


P(a,b;T1,T2/c:t) = P(a:T1/c:t)P(c;T3=t/b;t2=T2)/P(a;T1/b;t2=T2) (3)

Note that T3=t lies between T1 and T2=t2.


What will be of interest for us in the above relation is that one need not be able to write down psi(a,b;T1,T2/c;t) in order to calculate P(a,b;T1,T2/c;t). Each of the factors on the right-hand side of (3) is a fundamental probability, and these are calculated through (1), from the two-point propagators psi(a;T1/c;t), psi(c;t/b;T2), and psi(a;T1/b;T2) ... Given a complete set of two-point propagators, and nothing more, one can calculate a complete set of probabilities. (p.3318)
Can we define a complete set of commuting observables when we use the spacetime symmetry of the Lorentz group of special relativity? If not, we cannot define a state properly as the common eigenfunction of that set, and we cannot obey (iv) above, say the authors.


... a new and serious problem may arise here .. The source of the trouble is the requirement of relativistic causality, and the trouble is that this requirement certainly imposes new limitations on the kinds of measurement procedures that can be carried out.
The authors then do a gedankenexperiment in which is it supposed that nondemolition measurements are possible on a single-particle momentum eigenstate for Lorentz symmetry. They conclude that
Such a measurement, then, can move a particle around at superluminal velocities, and such a particle, or certainly an ensemble of such particles, can carry information between spacelike separated points, and this is a direct violation of the relativistic principle of causality. So this sort of nondemolition experiment is certainly impossible. p.3318
Therefore, relativistic causality gives rise to a new kind of uncertainty relation that the measurement of any state of spatial extension deltaX requires a measuring time deltaT of at least deltaX/c. See equations (5) and (6). However, they then show that this new uncertainty relation is not really correct.

One is able to measure nonlocal properties using only several local interactions. They use a two-particle system and a measuring apparatus. In the course of their difficult analysis we find


.. the full state cannot be separated into a two-particle state and a state of the measuring devices .. rather, the two systems are inexorably entangled here, and the interesting thing about this entanglement is that it is purely a product of the Lorentz transformation. In the old frame the two systems never get tangled at all, or more precisely the process of getting tangled and then getting untangled, which occur in the intervals t1 -> t2 and t3- >t4, respectively, in the new frame, are simultaneous in the old one.... In the old frame, then, this procedure, without disturbing in any way the state history in that frame, has changed the transformation properties of that history. So although the capacity of some experimental procedure to verify a given state is preserved under Lorentz transformations, the property of being a nondemolition experiment is not. This kind of procedure cannot monitor the history covariantly. p.3321
The authors then consider a nondemolition measurement on an EPR Bohm gedanken experiment on a pair of particles in a singlet spin 0 state using only local interactions. They extend this nondemolition analysis to a single-particle state which is a superposition of two localized but non-overlapping states in space. They find, consistent with causality,
So if, at the end of the procedure, we find a particle at x2, it is impossible to determine whether the particle has been moved there from x1 or created by the device at x2. There is not any means, then, of transferring information between x1 and x2 in this way ...

All of this is less than sufficient, however, to determine whether there are enough relativistic field-theoretic observables to monitor every possible system's history... But at least one general conclusion can be drawn, and that is that (5) and (6) are certainly not uncertainty principles; indeed we have seen explcitly how to violate them. p.3322

The key question for relativistic quantum mechanics is "how are the initial conditions for the propagator determined by experimental results?" The essential ambiguity is that different observers in relative motion have different spacelike surfaces of distant simultaneity. Therefore, if each such observer tries to use equation (2) above, they will get different frame-dependent probability distributions. This violates the requirement of relativistic causality "that local observables must commute at spacelike separations". p.3322 All of the different sets of initial conditions for different frames should produce identical probability distributions if causality is correct,
and indeed a measurement may be taken to impose initial conditions over any spacelike hypersurface containing the measurement event without altering these probabilities.
The authors conclude that in relativistic quantum mechanics (i) is obeyed but the idea of a state cannot be supported.


There are, then, no states at all which both transform properly and may be monitored in any frame. p. 3323
Indeed, the authors assert that (iii) and (iv) above contradict each other for the Lorentz group in contrast to the Galilei group where they are compatible.

 They also show that the postulate of invariant collapse along the backward light cone is not tenable because it violates the conservation of electric charge in a single frame. p.3324

 In summary, using the Bohr Copenhagen interpretation in which there are no hidden variable particles, it is not possible in special relativity with causality, to conceive of a quantum state in which a property has been measured to still have that property after the measurement even though it is not being disturbed.


... in this case the various elements of the definition of the state are mutually exclusive. A description of the physical system in terms of its observables simply cannot be consistently written down.
One way out is to violate causality. One can then have a state. One can also introduce the Bohm hidden variable which denies collapse and see what happens. I don't know the answer to the latter possibility.


Even if, say, one can predict with certainty that any measurement of the total charge of a system at any time in any frame will yield the value e, still neither this charge nor any other physical property may be consistently attributed to the state.

 The equations of motion and the postulate of collapse enter into the calculation of probabilities exactly as they do in the nonrelativistic case, but they can no longer be thought of as describing the evolution of the physical system, because it is impossible to define a consistent description of the system which collapses, or evolves in accordance with these equations.