Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

is appropriate that one of the very basic mathema-

tical aspects of the quantum formalism be addressed

first. It is an accepted aspect of the quantum

formalism that a state-vector such as j i should

not, in any case, be thought of as providing a unique

mathematical description of a ‘‘physical reality’’ for

the simple reason that j i and zj i, where z is any

nonzero complex number, describe precisely the

same physical situation. It is a common, but not

really necessary, practice to demand that j i be

normalized to unity: h j i= 1, in which case the

freedom in j i is reduced to the multiplication by a

phase factor j i7!e

i

j i. Either way, the physically

distinguishable states constitute a projective Hilbert

space PH, where each point of PH corresponds to a

one-dimensional linear subspace of the Hilbert space

H. The issue, therefore, is whether quantum reality

can be described in terms of the points of a

projective Hilbert space PH.

Reality in Spin-1/2 Systems

As a general comment, it seems that for systems with a

small number of degrees of freedom – that is, for a

Hilbert space H

of small finite dimension n –itseems

more reasonable to assign a reality to the elements of

than is the case when n is large. Let us begin with

a particularly simple case, where n = 2, and H

describes the two-dimensional space of spin states of

a massive particle of spin 1/2, such as an electron,

proton, or quark, or suitable atom. Here we can take

as an orthonormal pair of basis states jÆi and ji,

representing right-handed spin about the ‘‘up’’ and

‘‘down’’ directions, respectively. Clearly there is

nothing special about these particular directions, so

any other state of spin, of direction ji say, is just as

‘‘real’’ as the original two. Indeed, we always find

ji¼wjÆiþzji

for some pair of complex numbers z and w (not both

zero). The different possible ratios z : w give us a

complex plane (of zw

1

) compactified by a point at

infinity (where w = 0) – a ‘‘Riemann sphere’’ – which is

a realization of the complex projective 1-space PH

There does indeed seem to be something ‘‘real’’

about the spin state of such a spin-1/2 particle or

atom. We might imagine preparing the spin of

a suitable spin-1/2 atom using a Stern–Gerlach

apparatus (see Introductory Article: Quantum

Mechanics) oriented in some chosen direction. The

atom seems to ‘‘know ’’ the direction of its spin,

because if we measure it again in the same direction

it has to be prepared to give us the answer ‘‘YES,’’ to

the second measurement, with certainty, and that

direction for its spin state is the only one that can

guarantee this answer. (We are, of course, consider-

ing only ‘‘ideal’’ measurements, for the purpose of

argument.) Moreover, we could imagine that

between the two measurements, some appropriate

magnetic field had been introduced so as to rotate

the spin direction in some very specific way, so that

the spin state is now some other direction such as

jÇi. By rotating our second Stern–Gerlach apparatus

to agree with this new direction, we must again get

certainty for the YES answer, the guaranteeing of

this by the rotated state seeming now to give a

‘‘reality’’ to this new state jÇi. The quantum

formalism does not allow us to ascertain an

unknown direction of spin. But it does allow for us

to ‘‘confirm’’ (or ‘‘refute’’) a proposed direction for

the spin state, in the sense that if the proposed

direction is incorrect, then there is a nonzero

probability of refutation. Only the correct direction

can be guaranteed to give the YES answer.

EPR–Bohm and Bell’s Theorem

For a pair of particles or atoms of spin 1/2, the issue

of the ‘‘reality’’ of spin states becomes less clear.

Consider, for example, the EPR–Bohm example

(where ‘‘EPR’’ stands for Einstein–Podolski–Rosen)

whereby an initial state of spin 0 decays into two

spin-1/2 atoms, traveling in opposite directions (east

E, and west W). If a suitable Stern–Gerlach apparatus

is set up to measure the spin of the atom at E, finding

an answer jÇi, say, then this immediately ensures

that the state at W is the oppositely pointing jªi,

which can subsequently be ‘‘confirmed’’ by measure-

ment at W. This, then, seems to provide a ‘‘reality’’

for the spin state jªi at W as soon as the E

measurement has been performed, but not before.

Now, let us suppose that some orientation different

from ª had actually been set up for the measurement

at W, namely that which would have given YES for

the direction . This measurement can certainly give

the answer YES upon encountering jªi (with a

certain nonzero probability, namely (1 þ cos )=2,

where  is the angle between ª and ). So far, this

provides us with no problem with the ‘‘reality’’ of the

spin state of the atom at W, since it would have been

jªi before the measurement at W and would have

‘‘collapsed’’ (by the R-process) to ji after the

measurement. But now suppose that the measure-

ment at W had actually been performed momentarily

before the measurement at E, rather than just

after it. Then there is no reason that the

W-measurement would encounter jªi, rather than

some other direction, but the result ji of the

measurement at W now seems to force the state at

Etobeji. Indeed, the two measurements, at E and

262 Quantum Mechanics: Foundations

at W, might have been spacelike separated, and

because of the requirements of special relativity there

would be no meaning to say which of the two

measurements – at E or at W – had ‘‘actually’’

occurred first. One seems to obtain a different picture

of ‘‘reality’’ depending on this ordering.

In fact, the calculations of probab ilities come out

the same whichever picture is used, so if one asks

only for a calculational procedure for the probabil-

ities, rather than an actual picture of quantum

reality, these considerations are not problematic. But

they do provide profound diff iculties for any view of

quantum reality that is entirely local. The difficulty

is made particularly clear in a theorem due to John

Bell (1964, 1966a, b) which showed that on the

basis of the assumptions of local realism, there are

particular relations between the conditional prob-

abilities, whi ch must hold in any situation of this

kind; more over, these inequalities can be violated in

various situations in standard quantum mechanics.

(See, most specifically, Clauser et al. (1969).) Several

experiments that were subsequently performed

(notably Aspect et al. (1982)) confirmed the expec-

tations of quantum mechanics, thereby presenting

profound difficulties for any local realistic model of

the world. There are also situations of this kind

which involve only yes/no questions, so that actual

probabilities do not need to be considered, see

Kochen and Specker (1967), Peres (1991), Hardy

(1993), Conway and Kochen (2002 ). Basically: if

one insists on realism, then one must give up

locality. Moreover, nonlocal realistic models, con-

sistent with the requirements of special relativity, are

not easy to construct (see Quantum Mechanics:

Generalizations), and have so far proved elusive.

Other Aspects of Quantum Nonlocality

Problems of this kind occur even at the more

elementary level of single particles, if one tries to

consider that an ordinary particle wave function

(position-space description of j i) might be just

some kind of ‘‘local disturbance,’’ like an ordinary

classical wave. Consider the wave function spread-

ing out from a localized source, to be detec ted at a

perpendicular screen some distance away. The

detection of the pa rticle at any one place on the

screen immediately forbids the detection of that

particle at any other place on the screen, and if we

are to think of this information as being transmitted

as a classical signal to all other places on the screen,

then we are confronted with problems of super-

luminary communication. Again, any ‘‘realistic’’

picture of this process would require nonlocal

ingredients, which are difficult to square with the

requirements of special relativity. (It is possible that

these difficulties might be resolved within some kind

of nonlocal geometry, such as that supplied by

twistor theory (see Twistors; Twistor Theory: Some

Applications); see, particularly, Penrose (2005).)

These types of issues are made even more dramatic

and problematic in the procedure of ‘‘quantum

teleportation,’’ whereby the information in a quantum

state (e.g., the unknown actual direction  in some

quantum state ji) can be transported from one

experimenter A to another one B, by merely

the sending of a small finite number of classical bits

of information from A to B, where before this classical

information is transmitted, A and B must each be in

possession of one member of an EPR pair. More

explicitly, we may suppose A (Alice) is presented with

aspin-1/2stateji, but is not told the direction .She

has in her possession another spin-1/2 state which is an

EPR–Bohm partner of a spin-1/2 state in the posses-

sion of B (Bob). She combines this ji with her EPR

atom and then performs a measurement which

distinguishes the four orthogonal ‘‘Bell states’’

0: jÆijijijÆi

1: jÆijÆijiji

2: jÆijÆiþjiji

3: jÆijiþjijÆi

where the first state in each product refers to her

unknown state and the second refers to her EPR

atom. The result of this measurement is conveyed to

Bob by an ordinary classical signal, coded by the

indicated numbers 0, 1, 2, 3. On receiving Alice’s

message, Bob takes the other member of the EPR

pair and performs the following rotation on it:

0: leave alone

1: 180



about x-axis

2: 180



about y-axis

3: 180



about z-axis

This achie ves the successful ‘‘teleporting’’ of ji

from A to B, despite the fact that only 2 bits of

classical information have been signaled. It is the

acausal EPR–Bohm connection that provides the

transmission of ‘‘quantum information’’ in a classi-

cally acausal way. Again, we see the essentially

nonlocal (or acausal ) nature of any attempted

‘‘realistic’’ picture of quantum phenomena. It may

be regarded as inappropriate to use the term

‘‘information’’ for something that is propagated

acausally and cannot be directly used for signaling.

It has been suggested, accordingly, that a term such

as ‘‘quanglement’’ might be more appropriate to use

for this co ncept; see Penrose (2002, 2004).

Quantum Mechanics: Foundations 263

The preceding arguments illustrate how quantum

systems involving even just a few particles can exhibit

features quite unlike the ordinary behavior of classical

particles. This was pointed out by Schro¨dinger (1935),

and he referred to this key property of composite

quantum systems as ‘‘entanglement.’’ An entangled

quantum state (vector) is an element of a product

Hilbert space H

 H

which cannot be written as a

tensor product of elements j iji,withj i2H

and

ji2H

,whereH

refers to one part of the system and

refers to another part, usually taken to be physically

widely separated from the first. EPR systems are a

clear example, and we begin to see very nonclassical,

effectively nonlocal behavior with entangled systems

generally. A puzzling aspect of this is that the vast

majority of states are indeed entangled, and the more

parts that a system has, the more entangled it becomes

(where the generalization of this notion to more than

two parts is evident). One might have expected that

‘‘big’’ quantum systems with large numbers of parts

ought to behave more and more like classical systems

when they get larger and more complicated. However,

we see that this is very far from being the case. There is

no good reason why a large quantum system, left on its

own to evolve simply according to U should actually

resemble a classical system, except in very special

circumstances. Something of the nature of the R

process seems to be needed in order that classical

behaviour can ‘‘emerge.’’

Schro¨ dinger’s Cat

To clarify the nature of the problem we must consider a

key feature of the U formalism, namely ‘ ‘ linearity,’’

which is supposed to hold no matter how large or

complicated is the quantum system under considera-

tion. Recall the quantum superposition principle, which

allows us to construct arbitrary combinations of states

j i¼wjiþzji

from two given states ji and ji. Quantum linearity

tells us that if

ji

j

i and ji j

where the symbol ‘‘

’’ expresses how a state will

have evolved after a specified time period T, then

j i¼wjiþzji

i¼wj

iþzj

Let us now consider how this might be applied in

a particular, rather outlandish situation. Let us

suppose that the ji-evolution consists of a photon

going in one direction, encountering a detector,

which is connected to some murderous device which

kills a cat. The ji-evolution, on the other hand,

consists of the photon going in some other direction,

missing the detector so that the murderous device is

not activat ed, and the cat is left alive. These two

alternatives would each be perfectly plausible

evolutions which might take place in the physical

world. Now, by use of a beam splitter (effectively a

‘‘half-silvered mirror’’) we can easily arrange for the

initial state of the photon to be the superposition

wjiþzji of the two. Then by quantum linearity

we find, as the final result, the superposed state

wj

iþzj

i, in which the cat is in a superposition

of life and death (a ‘‘Schro¨ dinger’s cat’’).

We note that the two individual final states j

and j

i would each involve not just the cat but also

its environment, fully entangled with the cat’s state,

and perhaps also some human observer looking at

the cat. In the latter case, j

i would involve the

observer in a state of unhappily perceiving a dead

cat, and j

i happily perceiving a live one. Two of

the ‘‘conventional standpoints’’ with regard to the

measurement problem are of relevance here. Accord-

ing to the standpoint of environmental decoherence,

the details of the environmental degrees of freedom

are completely inaccessible, and it is deemed to be

appropriate to construct a density matrix to describe

the situation, which is a partial trace D of the

quantity j ih j, constructed by tracing out over all

the environmental degrees of freedom:

D ¼ trace over environmentfj ih jg

The density matrix tends to be regarded as a more

appropriate quantity than the ket jyi to represent

the physical situation, although this represents

something of an ‘‘ontology shift’’ from the point of

view that was being held previously. Under appro-

priate assum ptions, D may now be sho wn to attain a

form that is close to being diagonal in a basis with

respect to which the cat is either dead or alive, and

then, by a second ‘‘ontology shift’’ D is re-read as

describing a probability mixture of these two states.

According to the second ‘ ‘conventional standpoint’’

under consideration here, it is not logical to take this

detour through a density-matrix description, and

instead one should maintain a consistent ontology by

following the evolution of the state j i itself through-

out. The ‘‘real’’ resulting physical state is then taken to

be actually j

i, which involves the superposition of a

dead and live cat. Of course this ‘‘reality’’ does not agree

with the reality that we actually perceive, so the position

is taken that a conscious mind would not actually be

able to function in such a superposed condition, and

would have to settle into a state of perception of either a

dead cat or a live one, these two alternatives occurring

with probabilities as given by the Born rule stated

above. It may be argued that this conclusion depends

264 Quantum Mechanics: Foundations

upon some appropriate theory of how conscious minds

actually perceive things, and this appears to be lacking.

A good many physicists might argue that none of

these attempts at resolution of the measurement

problem is satisfactory, including ‘‘Copenhagen,’’

although the latter at least has the advantage of

offering a pragmatic, if not fully logical, stance. Such

physicists might take the position that it is necessary

to move away from the precise version of quantum

theory that we have at present, and turn to one of its

modifications. Some major candidates for modifica-

tion are discussed in Quantum Mechanics: General-

izations. Most of these actually make predictions

that, at some stage, would differ from those of

standard quantum mechanics. So it becomes an

experimental matter to ascertain the plausibility of

these schemes. In addition, there are reinterpretations

which do not change quantum theory’s predictions,

such as the de Broglie–Bohm model. In this, there are

two levels of ‘‘reality,’’ a firmer one with a particle or

position-space ontology, and a secondary one con-

taining waves which guide the behavior at the firmer

level. It is clear, however, that these issues will

remain the subject of debate for many years to come.

See also: Functional Integration in Quantum Physics;

Normal Forms and Semiclassical Approximation;

Quantum Mechanics: Generalizations; Twistor Theory:

Some Applications [In Integrable Systems, Complex

Geometry and String Theory]; Twistors.

Further Reading

Aspect A, Grangier P, and Roger G (1982) Experimental

realization of Einstein–Podolsky–Rosen–Bohm. Gedankenex-

periment: a new violation of Bell’s inequalities. Physical

Review Letters 48: 91–94.

Bell JS (1964) On the Einstein Podolsky Rosen paradox. Physics

1: 195–200. Reprinted in Quantum Theory and Measurement,

eds., Wheeler JA and Zurek WH (Princeton Univ. Press,

Princeton, 1983).

Bell JS (1966a) On the problem of hidden variables in quantum

theory. Reviews in Modern Physics 38: 447–452.

Bell JS (1966b) Speakable and Unspeakable in Quantum Mechanics.

Cambridge: Cambridge University Press. Reprint 1987.

Clauser JF, Horne MA, Shimony A, and Holt RA (1969)

Proposed experiment to test local hidden-variable theories.

Physical Review Letters 23: 880–884.

Conway J and Kochen S (2002) The geometry of the quantum

paradoxes. In: Bertlmann RA and Zeilinger A (eds.) Quantum

[Un]speakables: From Bell to Quantum Information, Ch. 18

(ISBN 3-540-42756-2). Berlin: Springer.

Hardy L (1993) Nonlocality for two particles without inequalities

for almost all entangled states. Physical Review Letters

71(11): 1665.

Kochen S and Specker EP (1967) Journal of Mathematics and

Mechanics 17: 59.

Penrose R (2002) John Bell, state reduction, and quanglement. In:

Reinhold A, Bertlmann, and Zeilinger A (eds.) Quantum

[Un]speakables: From Bell to Quantum Information,

pp. 319–331. Berlin: Springer.

Penrose R (2004) The Road to Reality: A Complete Guide to the

Laws of the Universe. London: Jonathan Cape.

Penrose R (2005) The twistor approach to space-time struc-

tures. In: Ashtekar A (ed.) In 100 Years of Relativity; Space-

Time Structure: Einstein and Beyond. Singapore: World

Scientific.

Peres A (1991) Two simple proofs of the Kochen–Specker

theorem. Journal of Physics A: Mathematical and General

24: L175–L178.

Schro¨ dinger E (1935) Probability relations between separated

systems. Proceedings of the Cambridge Philosophical Society

31: 555–563.

von Neumann J (1932) Mathematische Grundlageen der Quan-

tenmechanik. Berlin: Springer.

von Neumann J (1955) Mathematical Foundations of Quantum

Mechanics. Princeton: Princeton University Press.

Quantum Mechanics: Generalizations

P Pearle, Hamilton College, Clinton, NY, USA

A Valentini, Perimeter Institute for Theoretical

Physics, Waterloo, ON, Canada

Introduction

According to the so-called ‘ ‘Copenhagen Interpreta-

tion,’’ standard quantum theory is limited to describ-

ing experimental situations. It is at once remarkably

successful in its predictions, and remarkably ill-defined

in its conceptual structure: what is an experiment?

what physical objects do or do not require

quantization? how are the states realized in nature to

be characterized? how and when is the wave-function

‘‘collapse postulate’’ to be invoked? Because of its

success, one may suspect that quantum theory can be

promoted from a theory of measurement to a theory

of reality. But, that requires there to be an unambig-

uous specification (S) of the possible real states of

nature and their probabilities of being realized.

There are several approaches that attempt to

achieve S. The more conservative approaches (e.g.,

consistent histories, environmental decoherence,

many worlds) do not produce any predictions that

differ from the standard ones because they do not

tamper with the usual basic mathematical

Quantum Mechanics: Generalizations 265

formalism. Rather, they utilize structures compatible

with standard quantum theory to elucidate S. These

approaches, which will not be discussed in this

article, have arguably been less successful so far at

achieving S than approaches that introduce

significant alterations to quantum theory.

This article will largely deal with the two most

well-developed realistic models that reproduce

quantum theory in some limit and yield potentially

new and testable physics outside that limit. First, the

pilot-wave model, which will be discussed in the

broader context of ‘‘hidden-variables theories.’’

Second, the continuous spontaneous localization

(CSL) model, which describes wave-function col-

lapse as a physical process. Other related models

will also be discussed briefly.

Due to bibliographic space limitations, this article

contains a number of uncited references, of the form

‘‘[author] in [year].’’ Those in the next section can

be found in Valentini (2002b, 2004a,b) or at

www.arxiv.org. Those in the subsequent sections

can be found in Adler (2004), Bassi and Ghirardi

(2003), Pearle (1999) (or in subsequent papers by

these authors, or directly, at www.arxiv.org), and in

Wallstrom (1994).

Hidden Variables and Quantum

Nonequilibrium

A deterministic hidden-variables theory defines a

mapping ! = !(M, ) from initial hidden parameters

 (defined, e.g., at the time of preparation of a

quantum state) to final outcomes ! of quantum

measurements. The mapping depends on macro-

scopic experimental settings M, and fixes the out-

come for each run of the experiment. Bell’s theorem

of 1964 shows that, for entangled quantum states of

widely separated systems, the mapping must be

nonlocal: some outcomes for (at least) one system

must depend on the setting for another distant

system.

In a viable theory, the statistics of quantum

measurement outcomes – over an ensemble of

experimental trials with fixed settings M – will

agree with quantum theory for some special dis-

tribution 

() of hidden variables. For example,

expectation values will coincide with the predictions

of the Born rule

h!i



d

ðÞ!ðM;Þ¼trð



Þ

for an appropriate density operator

 and Hermi-

tian observable

. (As is customary in this context,

d is to be understood as a generalized sum.)

However, given the mapping ! = !(M, ) for indi-

vidual trials, one may, in principle, consider

nonstandard distributions () 6¼ 

()thatyield

statistics outside the domain of ordinary quantum

theory (Valentini 1991, 2002a). We may say that

such distributions correspond to a state of quantum

nonequilibrium.

Quantum nonequilibrium is characterized by the

breakdown of a number of basic quantum con-

straints. In particular, nonlocal signals appear at the

statistical level. We shall first illustrate this for the

hidden-variables model of de Broglie and Bohm.

Then we shall generalize the discussion to all

(deterministic) hidden-variables theories.

At present there is no experimental evidence for

quantum nonequilibrium in nature. However, from

a hidden-variables perspective, it is natural to

explore the theoretical properties of nonequilibrium

distributions, and to search experimentally for the

statistical anomalies associated with them.

From this point of view, quantum theory is a

special case of a wider physics, much as thermal

physics is a special case of a wider (nonequilibrium)

physics. (The special distribution 

() is analo-

gous to, say, Maxwell’s distribution of molecular

speeds.) Quantum physics may be compared with

the physics of global thermal equilibrium, which is

characterized by constraints – such as the impossi-

bility of converting heat into work (in the absence of

temperature differences) – that are not fundamental

but contingent on the state. Similarly, quantum

constraints such as statistical locality (the impossi-

bility of converting entanglement into a practical

signal) are seen as contingencies of 

().

Pilot-Wave Theory

The de Broglie–Bohm ‘‘pilot-wave theory’’ – as it

was originally called by de Broglie, who first

presented it at the Fifth Solvay Congress in 1927 –

is the classic example of a deterministic hidden-

variables theory of broad scope (Bohm 1952, Bell

1987, Holland 1993). We shall use it to illustrate the

above ideas. Later, the discussion will be generalized

to arbitrary theories.

In pilot-wave dynamics, an individual closed

system with (configuration-space) wave function

(X, t) satisfying the Schro¨ dinger equation

ih

@

H ½1

has an actual configuration X(t) with velocity

XðtÞ¼

JðX ; tÞ

jðX; tÞj

½2

266 Quantum Mechanics: Generalizations

where J = J[] = J(X, t) satisfies the continuity

equation

@jj

þrJ ¼ 0 ½3

(which follows from [1]). In quantum theory, J is the

‘‘probability current.’’ In pilot-wave theory,  is an

objective physical field (on configuration space)

guiding the motion of an individual system.

Here, the objective state (or ontology) for a closed

system is given by  and X. A probability distribu-

tion for X – discussed below – completes an

unambiguous specification S (as mentioned in the

introduction).

Pilot-wave dynamics may be applied to any

quantum system with a locally conserved current in

configuration space. Thus, X may represent a many-

body system, or the configuration of a continuous

field, or perhaps some other entity.

For example, at low energies, for a system of N

particles with positions x

(t) and masses

(i = 1, 2, ..., N), with an external potential V,

[1] (with X  (x

, x

, ..., x

)) reads

ih

@

i¼1



h

 þ V ½4

while [2] has components

h





½5

(where =jje

(i=h)S

In general, [1] and [2] determine X(t) for an

individual system, given the initial conditions

X(0),  (X,0) at t = 0. For an arbitrary initial

distribution P(X, 0), over an ensemble with the

same wave function (X, 0), the evolution P(X, t)

of the distribution is given by the continuity

equation

þrðP

XÞ¼0 ½6

The outcome of an experiment is determined by

X(0),  (X, 0), which may be identified with . For

an ensemble with the same (X, 0), we have

 = X(0).

Quantum equilibrium From [3] and [6],ifwe

assume P(X,0)= j (X,0)j

at t = 0, we obtain

P(X, t) = j(X, t)j

– the Born-rule distribution of

configurations – at all times t.

Quantum measurements are, like any other

process, described and explained in terms of evol-

ving configurations. For measurement devices whose

pointer readings reduce to configurations, the

distribution of outcomes of quantum measurements

will match the statistical predictions of quantum

theory (Bohm 1952, Bell 1987, Du¨rr et al. 2003).

Thus, quantum theory emerges phenomenologically

for a ‘‘quantum equilibrium’’ ensemble with

distribution P(X, t) = j(X, t)j

(or () = 

()).

Quantum nonequilibrium In principle, as we saw

for general hidden-variables theories, we may con-

sider a nonequilibrium distribution P(X,0)6¼

j(X,0)j

of initial configurations while retaining

the same deterministic dynamics [1], [2] for indivi-

dual systems (Valentini 1991). The time evolution of

P(X, t) will be determined by [6].

As we shall see, in appropriate circumstances

(with a sufficiently complicated velocity field

X), [6]

generates relaxation P !jj

on a coarse-grained

level, much as the analogous classical evolution on

phase space generates thermal relaxation. But for as

long as the ensemble is in nonequilibrium, the

statistics of outcomes of quantum measurements

will disagree with quantum theory.

Quantum nonequilibrium may have existed in the

very early universe, with relaxation to equilibrium

occurring soon after the big bang. Thus, a hidden-

variables analog of the classical thermodynamic

‘‘heat death of the universe’’ may have actually

taken place (Valentini 1991). Even so, relic cosmo-

logical particles that decoupled sufficiently early

could still be in nonequilibrium today, as suggested

by Valentini in 1996 and 2001. It has also been

speculated that nonequilibrium could be generated

in systems entangled with degrees of freedom behind

a black-hole event horizon (Valentini 2004a).

Experimental searches for nonequilibrium have

been proposed. Nonequilibrium could be detected

by the statistical analysis of random samples of

particles taken from a parent population of (for

example) relics from the early universe. Once the

parent distribution is known, the rest of the popula-

tion could be used as a resource, to perform tasks

that are currently impossible (Valentini 2002b).

H-Theorem: Relaxation to Equilibrium

Before discussing the potential uses of nonequili-

brium, we should first explain why all systems

probed so far have been found in the equilibrium

state P = jj

. This distribution may be accounted

for along the lines of classical statistical mechanics,

noting that all currently accessible systems have had

a long and violent astrophysical history.

Dividing configuration space into small cells, and

introducing coarse-grained quantities



jj

, a gen-

eral argument for relaxation



P !

jj

is based on an

Quantum Mechanics: Generalizations 267

analog of the classical coarse-graining H-theorem.

The coarse-grained H-function



H ¼



P lnð



jj

Þ½7

(minus the relative entropy of



P with respect to

jj

) obeys the H-theorem (Valentini 1991)



HðtÞ



Hð0Þ

(assuming no initial fine-grained microstructure in P

and jj

). Here,



H  0 for all



P, j j

and



H = 0if

and only if



P =

jj

everywhere.

The H-theorem expresses the fact that P and jj

behave like two ‘‘fluids’’ that are ‘‘stirred’’ by the same

velocity field

X, so that P and jj

tend to become

indistinguishable on a coarse-grained level. Like its

classical analog, the theorem provides a general

understanding of how equilibrium is approached,

while not proving that equilibrium is actually

reached. (And of course, for some simple systems –

such as a particle in the ground state of a box, for

which the velocity field rS=m vanishes – there is no

relaxation at all.) A strict decrease of



H(t) immedi-

ately after t = 0 is guaranteed if

r(P

=j

) has

nonzero spatial variance over a coarse-graining cell,

as shown by Valentini in 1992 and 2001.

A relaxation timescale  may be defined by

1=

(d



H=dt

)



. For a single particle with

quantum energy spread E, a crude estimate given

by Valentini in 2001 yields   (1=")h

1=2

(E)

3=2

where " is the coarse-graining length. For wave

functions that are superpositions of many energy

eigenfunctions, the velocity field (generally) varies

rapidly, and detailed numerical simulations (in two

dimensions) show that relaxation occurs with an

approximately exponential decay



H(t) 



t=t

with a time constant t

of order  (Valentini and

Westman 2005).

Equilibrium is then to be expected for particles

emerging from the violence of the big bang. The

possibility is still open that relics from very early

times may not have reached equilibrium before

decoupling.

Nonlocal Signaling

We now show how nonequilibrium, if it were ever

discovered, could be used for nonlocal signaling.

Pilot-wave dynamics is nonlocal. For a pair of

particles A, B with entangled wave function

(x

, x

, t), the velocity

(t) = r

S(x

, x

, t)=m

of A depends instantaneously on x

, and local

operations at B – such as switching on a potential –

instantaneously affect the motion of A. For an

ensemble P(x

, x

, t) = j(x

, x

, t)j

, local opera-

tions at B have no statistical effect at A: the

individual nonlocal effects vanish upon averaging

over an equilibrium ensemble.

Nonlocality is (generally) hidden by statistical

noise only in quantum equilibrium. If instead

P(x

, x

,0)6¼j(x

, x

,0)j

, a local change in the

Hamiltonian at B generally induces an instan-

taneous change in the marginal p

, t) 

P(x

, x

, t)atA. For example, in one dimen-

sion a sudden change

in the Hamiltonian

at B induces a change p

 p

, t)  p

,0)

(for small t)(Valentini 1991),

p

¼



aðx

bðx



Pðx

; x

; 0Þjðx

; x

; 0Þj

jðx

; x

; 0Þj



½8

(Here m

= m

= m, a(x

) depends on (x

, x

, 0),

while b(x

) also depends on

and vanishes if

.) The signal is generally nonzero if

6¼j

Nonlocal signals do not lead to causal paradoxes

if, at the hidden-variable level, there is a preferred

foliation of spacetime with a time parameter that

defines a fundamental causal sequence. Such sig-

nals, if they were observed, would define an

absolute simultaneity as discussed by Valentini in

1992 and 2005. Note that in pilot-wave field

theory, Lorentz invariance emerges as a phenom-

enological symmetry of the equilibrium state,

conditional on the structure of the field-theoretical

Hamiltonian (as discussed by Bohm and Hiley in

1984, Bohm, Hiley and Kaloyerou in 1987, and

Valentini in 1992 and 1996).

Subquantum Measurement

In principle, nonequilibrium particles could also be

used to perform ‘‘subquantum measurements’’ on

ordinary, equilibrium systems. We illustrate this

with an exactly solvable one-dimensional model

(Valentini 2002b).

Consider an apparatus ‘‘pointer’’ coordinate y,

with known wave function g

(y) and known

(ensemble) distribution 

(y) 6¼ g

(y)

, where 

(y)

has been deduced by statistical analysis of random

samples from a parent population with known wave

function g

(y). (We assume that relaxation may be

neglected: for example, if g

is a box ground state,

y = 0 and 

(y) is static.) Consider also a ‘‘system’’

coordinate x with known wave function

(x) and

known distribution 

(x) =

(x)

.If

(y)is

arbitrarily narrow, x

can be measured without

268 Quantum Mechanics: Generalizations

disturbing

(x), to arbitrary accuracy (violating the

uncertainty principle).

To do this, at t = 0 we switch on an interaction

Hamiltonian

H = a

, where a is a constant and p

is canonically conjugate to y. For relatively large a,

we may neglect the Hamiltonians of x and y.For

=(x, y, t), we then have @=@t = ax@=@y.

For 

we have the continuity equation @j j

=@t =

ax@ 

=@y, which implies the hidden-variable

velocity fields

x = 0,

y = ax and trajectories x(t) = x

y(t) = y

þ ax

The initial product 

(x, y) =

(x)g

(y) evolves

into (x, y, t) =

(x)g

(y  axt). For at !0 (with a

large but fixed), (x, y, t) !

(x)g

(y) and

(x)is

undisturbed: for small at, a standard quantum

pointer with the coordinate y would yield negligible

information about x

. Yet, for arbitrarily small at,

the hidden-variable pointer coordinate y(t) = y

þ ax

does contain complete information about x

(and

x(t) = x

). This ‘‘subquantum’’ information will be

visible to us if 

(y) is sufficiently narrow.

For, over an ensemble of similar experiments,

with initial joint distribution P

(x, y) =

(x)



(y)

(equilibrium for x and nonequilibrium for y), the

continuity equation @P=@t =ax@P=@y implies that

P(x, y, t) =

(x)



(y  axt). If 

(y) is localized

around y = 0(

(y) = 0 for jyj > w=2), then a stan-

dard (faithful) measurement of y with result y

meas

will imply that x lies in the interval (y

meas

=at  w=2at,

meas

=at þ w=2at) (so that P(x, y, t) 6¼ 0). Taking the

simultaneous limits at !0, w !0, with w=at !0,

the midpoint y

meas

=at !x

(since y

meas

= y

þ ax

and y

 w=2), while the error w=2at !0.

If w is arbitrarily small, a sequence of such

measurements will determine the hidden trajectory

x(t) without disturbing (x, t), to arbitrary accuracy.

Subquantum Information and Computation

From a hidden-variables perspective, immense phy-

sical resources are hidden from us by equilibrium

statistical noise. Quantum nonequilibrium would

probably be as useful technologically as thermal or

chemical nonequilibrium.

Distinguishing nonorthogonal states In quantum

theory, nonorthogonal states

ji,

ji(h

i6¼ 0)

cannot be distinguished without disturbing them.

This theorem breaks down in quantum nonequili-

brium (Valentini 2002b). For example, if

are distinct states of a single spinless particle, then

the associated de Broglie–Bohm velocity fields will

in general be different, even if h

i 6¼ 0, and so

will the hidden-variable trajectories. Subquantum

measurement of the trajectories could then distin-

guish the states j

i, j

Breaking quantum cryptography The security of

standard protocols for quantum key distribution

depends on the validity of the laws of quantum

theory. These protocols would become insecure

given the availability of nonequilibrium systems

(Valentini 2002b).

The protocols known as BB84 and B92 depend on

the impossibility of distinguishing nonorthogonal

quantum states without disturbing them. An eaves-

dropper in possession of nonequilibrium particles could

distinguish the nonorthogonal states being transmitted

between two parties, and so read the supposedly secret

key. Further, if subquantum measurements allow an

eavesdropper to predict quantum measurement out-

comes at each ‘‘wing’’ of a (bipartite) entangled state,

then the EPR (Einstein–Podolsky–Rosen) protocol also

becomes insecure.

Subquantum computation It has been suggested

that nonequilibrium physics would be computation-

ally more powerful than quantum theory, because of

the ability to distinguish nonorthogonal states

(Valentini 2002b). However, this ability depends

on the (less-than-quantum) dispersion w of the

nonequilibrium ensemble. A well-defined model of

computational complexity requires that the

resources be quantified in some way. Here, a key

question is how the required w scales with the size

of the computational task. So far, no rigorous results

are known.

Extension to All Deterministic

Hidden-Variables Theories

Let us now discuss arbitrary (deterministic) theories.

Nonlocal signaling Consider a pair of two-state

quantum systems A and B, which are widely

separated and in the singlet state. Quantum

measurements of observables



 m









(where m

, m

are unit vectors in Bloch

space and

are Pauli spin operators) yield

outcomes 

, 

= 1, in the ratio 1 : 1 at each

wing, with a correlation



= m

 m

. Bell’s

theorem shows that for a hidden-variables theory to

reproduce this correlation – upon averaging over an

equilibrium ensemble with distribution 

()–it

must take the nonlocal form



¼ 

ðm

; m

;Þ;

¼ 

ðm

; m

;Þ½9

More precisely, to obtain 



= m

 m

(where 





d

()



), at least one of

Quantum Mechanics: Generalizations 269



, 

must depend on the measurement setting at

the distant wing. Without loss of generality, we

assume that 

depends on m

For an arbitrary nonequilibrium ensemble with

distribution () 6¼ 

(), in general 





d()



differs from m

 m

,andtheout-

comes 

, 

= 1 occur in a ratio different from 1 : 1.

Further, a change of setting m

at B will generally

induce a change in the outcome statistics at A, yielding a

nonlocal signal at the statistical level. To see this, note

that, in a nonlocal theory, the ‘‘transition sets’’

ð; þÞ  j

ðm

; m

;Þ

¼1;



ðm

; m

;Þ¼þ1

ðþ; Þ  j

ðm

; m

;Þ

¼þ1;



ðm

; m

;Þ¼1

cannot be empty for arbitrary settings. Yet, in quantum

equilibrium, the outcomes 

= 1 occur in the ratio

1 : 1 for all settings, so the transition sets must

have equal equilibrium measure, 

(, þ)] =



(þ,)] (d

 

()d). That is, the

fraction of the equilibrium ensemble making the

transition 

= 1 !

= þ1 under m

must

equal the fraction making the reverse transition



= þ1 !

= 1. (This ‘‘detailed balancing’’ is

analogous to the principle of detailed balance in

statistical mechanics.) Since T

(, þ), T

(þ, )are

fixed by the deterministic mapping, they are indepen-

dent of the ensemble distribution (). Thus, for

() 6¼

(), in general [T

(, þ)] 6¼[T

(þ, )]

(d ()d): the fraction of the nonequilibrium

ensemble making the transition 

= 1!

= þ1

will not in general balance the fraction making the

reverse transition. The outcome ratio at A will then

change under m

and there will be an instanta-

neous signal at the statistical level from B to A

(Valentini 2002a).

Thus, in any deterministic hidden-variables

theory, nonequilibrium distributions () 6¼ 

()

generally allow entanglement to be used for non-

local signalling (just as, in ordinary statistical

physics, differences of temperature make it possible

to convert heat into work).

Experimental signature of nonequilibrium Quantum

expectations are additive, hc



þ c



i= c



iþ



i, even for noncommuting observables

([



] 6¼ 0, with c

, c

real). As emphasized by

Bell in 1966, this seemingly trivial consequence

of the (linearity of the) Born rule h

i= tr(



)is

remarkable because it relates statistics from

distinct, ‘‘incompatible’’ experiments. In none-

quilibrium, such additivity generically breaks

down (Valentini 2004b).

Further, for a two-state system with observables

m 

s, the ‘‘dot-product’’ structure of the quantum

expectation m 

shi= tr(

m 

s) = m  P (for some

Bloch vector P) is equivalent to expectation

additivity (Valentini 2004b). Nonadditive expecta-

tions then provide a convenient signature of none-

quilibrium for any two-state system. For example,

the sinusoidal modulation of the quantum trans-

mission probability for a single photon through a

polarizer

ðÞ¼

1 þ m 

ðÞ¼

1 þ P cos 2ðÞ½10

(where an angle  on the Bloch sphere corresponds

to a physical angle ==2) will generically break

down in nonequilibrium. Deviations from [10]

would provide an unambiguous violation of quan-

tum theory (Valentini 2004b).

Such deviations were searched for by Papaliolios

in 1967, using laboratory photons and successive

polarization measurements over very short times, to

test a hidden-variables theory (distinct from pilot-

wave theory) due to Bohm and Bub (1966), in which

quantum measurements generate nonequilibrium for

short times. Experimentally, successive measure-

ments over timescales 10

13

s agreed with the

(quantum) sinusoidal modulation cos

 to <



1%.

Similar tests might be performed with photons of a

more exotic origin.

Continuous Spontaneous Localization

Model (CSL)

The basic postulate of CSL is that the state vector

j , ti represents reality. Since, for example, in

describing a measurement, the usual Schro¨ dinger

evolution readily takes a real state into a nonreal

state, that is, into a superposition of real states

(such as apparatus states describing different

experimental outcomes), CSL requires a modifica-

tion of Schro¨ dinger’s evolution. To the Hamiltonian

is added a term which depends upon a classical

randomly fluctuating field w(x, t) and a mass-

density operator

A(x, t). This term acts to collapse

a superposition of states, which differ in their

spatial distribution of mass density, to one of these

states. The rate of collapse is very slow for a

superposition involving a few particles, but very

fast for a superposition of macroscopically different

states. Thus, very rapidly, what you see (in nature)

is what you get (from the theory). Each state vector

evolving under each w(x, t) corresponds to a

realizable state, and a rule is given for how to

associate a probability with each. In this way, an

270 Quantum Mechanics: Generalizations

unambiguous specification S, as mentioned in the

introduction, is achieved.

Requirements for Stochastic Collapse Dynamics

Consider a normalized state vector j , ti=



(t)ja

i(ha

i= 

) which undergoes a

stochastic dynamical collapse process. This means

that, starting from the initial superposition at t = 0,

for each run of the process, the squared amplitudes

(t) j

(t)j

fluctuate until all but one vanish, that

is, x

(1) = 1, (x

6¼m

(1) = 0) with probability x

(0).

This may be achieved simply, assuming negligible

effect of the usual Schro¨ dinger evolution, if the

stochastic process enjoys the following properties

(Pearle 1979):

ðtÞ¼1 ½11a

ðtÞ¼x

ð0Þ½11b

ð1Þx

ð1Þ ¼ 0 for m 6¼ n ½11c

where the overbar indicates the ensemble average at

the indicated time. The only way that a sum of

products of non-negative terms can vanish is for at

least one term in each product to vanish. Thus,

according to [11c], for each run, at least one of each

pair {x

(1), x

(1)}(n 6¼ m) must vanish. This

means that at most one x

(1) might not vanish

and, by [11a], applied at t = 1, one x

(1) must not

vanish and, in fact, must equal 1: hence, each run

produces collapse. Now, let the probability of the

outcome {x

(1) = 1, x

6¼n

(1) = 0} be denoted P

.Since

(1) = 1  P

m6¼n

0  P

= P

then, according to

the Martingale property [11b], applied at

t = 1, P

= x

(0): hence, the ensemble of runs pro-

duces the probability postulated by the usual ‘‘collapse

rule’’ of standard quantum theory.

A (nonquantum) stochastic process which obeys

these equations is the gambler’s ruin game. Suppose

one gambler initially possesses the fraction x

(0) of

their joint wealth, and the other has the fraction

(0). They toss a coin: heads, a dollar goes from

gambler 1 to gambler 2, tails the dollar goes the

other way. [11a] is satisfied since the sum of money

in the game remains constant, [11b] holds because it

is a fair game, and [11c] holds because each game

eventually ends. Thus, gambler i wins all the money

with probability x

(0).

CSL in Essence

Consider the (nonunitary) Schro¨ dinger picture evo-

lution equation

j ; ti

¼T exp 

þð4Þ

1

½wðt

Þ2

A

j ; 0i½12

where

H is the usual Hamiltonian, w(t

)isan

arbitrary function of white noise class,

A is a

Hermitian operator (

Aja

i= a

i),  is a collapse

rate parameter, T is the time-ordering operator and

h = 1. Associated with this, the probability rule

ðwÞDw 

h ; tj ; ti

t=dt

j¼0

dwðt

Þ=ð2=dtÞ

1=2

½13

is defined, which gives the probability that nature

chooses a noise which lies in the range {w(t

), w(t

) þ

dw(t

)} for 0  t

 t (for calculational purposes,

time is discretized, with t

= 0).

Equations [12] and [13] contain the essential

features of CSL, and are all that is needed to discuss

the simplest collapse behavior. Set

H = 0, so there is

no competition between collapse and the usual

Schro¨ dinger evolution, and let the initial state vector

be j ,0i=



i. Equations [12] and [13]

become

j ; ti



iexp



ð4Þ

1

½wðt

2a



½14a

ðwÞ¼

j

exp



ð2Þ

1

½wðt

 2a



½14b

When the unnormalized state vector in [14a] is

divided by P

1=2

(w) and so normalized, the squared

amplitudes are

ðtÞ¼j

exp ð2Þ

1





½wðt

Þ2a



ðwÞ

which are readily shown to satisfy [11a], [11b], and

[11c] in the form

1=2

(1)x

1=2

(1)= 0(m 6¼ n) (which

does not change the argument in the last subsection,

but makes for an easier calculation). Thus, [14a] and

[14b] describe collapse dynamics.

Quantum Mechanics: Generalizations 271