Are there quantum states?

Date: Thu, 2 Nov 1995 17:27:57 -0800
From: Jack Sarfatti <sarfatti@ix.netcom.com>
Reply to: quantum-d@teleport.com
To: quantum-d@teleport.com
Subject: QUANTUM-D: Are there quantum states?

Comments on Aharonov and Albert's "States and observables 
in relativistic quantum field theories," Phys. Rev. D, 21 
p. 3316, June 1980

This note examines a paper by Aharonov and Albert which
points out the breakdown of the usual the idea of the quantum 
state in relativistic quantum mechanics. If one assumes the 
usual idea of causality in relativistic quantum field theory, 
then there is no way to consistently describe the nonunitary 
collapse of a quantum state in contrast to the situation in 
nonrelativistic quantum mechanics. One possibility is to avoid 
the nonunitarities of wavefunction collapse which are so
problematic by instead accepting the explicitly nonlocal 
ontology of Bohm's theory. 

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. 

I. States and observables

In what sense do we meaningfully speak of states?

 "Any nonrelativistic quantum-mechanical system
 may at any time in its history be associated with  
 well-defined values of some complete commuting 
 subset of the observables of that system, and these
 values may always be verified directly by experiment."

For example, if an isolated physical system is found
to have a total momentum P, and if the system is
not subsequently disturbed then we describe the 
system as being in a state of momentum P even when
it is not in the process of being observed. Of course
if we go and check then we will discover that indeed
the momentum is P.

Aharonov and Albert, working implicitly within the 
Copenhagen framework, are investigating whether 
these simple criteria of "state" are met in the case
of relativistic field theories. 

They remark that

"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 [our main 
 result] is that although relativistic field theories 
 have the former capacity, they lack the latter one."

The authors show that "various nonlocal properties 
of certain systems" can be directly measured but 
are nevertheless not suitable to support the idea 
of a quantum state.

"We will study a simple nonlocal system for which
 a complete communting subset of measureable 
 observables exists, and consider whether the values
 of these observables may be incorporated into a
 covariant definition of the state of this system, 
 and will show that in fact no such state can be
 constructed, because no relativistically satisfactory
 version of the collapse-postulate can be found.
 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."

Definition

      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. 

Desirata - 

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

 (ii) Also, 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), then 
      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."

The authors use (iv) to approach the usual idea
of a state, as follows. 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." 

All of these desirata are obeyed for the Galilean 
group of nonrelativistic quantum mechanics. In
the nonrelativistic theory (i), (ii), and (iii) are 
consistent with (iv).

"So in norelativistic quantum mechanics one can
 write down a complete history of the physical
 system, even for times when the system is not 
 being observed, which transforms and evolves
 the appropriate ways, and can be fully verified
 by experiment."

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)

So much for the nonrelativistic case. What about the
relativistic case? Can we define a complete set of 
commuting observables when we use the spacetime 
symmetry of the Lorentz group of special relativity? 

If not then 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."

In order to investigate the limitations imposed
by relativity on measurement, the authors do a
gedankenexperiment in which it is supposed that 
nondemolition measurements are possible on a 
single-particle momentum eigenstate... Such a
momentum state measurement "instantaneously 
spreads the probability uniformally all over 
space." They conclude that 

"Such a measurement, then, can move a particle 
 around at superluminal velocities, and such a 
 particle, or 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.
     In the past it has been thought that this
 represents a restriction on the set of quantum-
 mechanical observables..." p.3318

However, the authors show that despite the new
limits imposed by relativity, "various nonlocal 
properties of certain systems" can be directly 
measured. One is able to measure such nonlocal 
properties using only a combination of local 
interactions.  On the other hand, it turns out 
that the these nonlocal measurements cannot be 
nondemolition (monitoring) measurements of the 
kind used in the nonrelativistic case to justify 
the concept of quantum state. 

The authors investigate 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. We will show that indeed 
 there can be in principle no process of any kind 
 capable of covariantly monitoring such a state
 history." p.3321

The authors also investigate some properties which 
_can_ be covariantly monitored. They 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 
find, consistent with causality, that based on
the possible monitorable properties of the system

"... 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, and thus there 
 is no violation of causality." p.3322

II. Are there quantum states? 

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 diferent 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) 
contradict each other for the Lorentz group in 
contrast to the Galilei group where they are 
compatible.

In summary, using Bohr's 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.

 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."

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.

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