Griffiths R.B. Consistent Quantum Theory

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354 Decoherence and the classical limit

The reduced density matrix ρ

for the particle just before it passes through the

second beam splitter is again of the form (26.8) or (26.9), with

α =#





=α

···α

, (26.15)

where

=#





, (26.16)

and ρ

, when the particle has passed through the second beam splitter, is again

given by (26.11).

In a typical situation one would expect the α

to be less than 1, though not

necessarily small. Note that if there are a large number of collisions, α in (26.15)

can be very small, even if the individual α

are not themselves small quantities.

Thus repeated interactions with the environment will in general lead to greater

decoherence than that produced by a single interaction, and if these interactions are

of roughly the same kind, one expects the coherence |α| to decrease exponentially

with the number of interactions.

Even if the different interactions with the environment are not independent of

one another, the net effect may well be much the same, although it might take more

interactions to produce a given reduction of |α|. In any case, what happens at the

second beam splitter, in particular the probability that the particle will emerge in

each of the output channels, depends only on the density matrix ρ for the particle

when it arrives at this beam splitter, and not on the details of all the scattering

processes which have occurred earlier. For this reason, a density matrix is very

convenient for analyzing the nature and extent of decoherence in this situation.

26.4 Random environment

Suppose that the environment which interacts with the particle is itself random, and

that at time t

, when the particle emerges from the ﬁrst beam splitter, it is described

by a density matrix R

which can be written in the form



#

|, (26.17)

where {|#

} is an orthonormal basis of E,and



= 1. Although it is natural,

and for many purposes not misleading to think of the environment as being in the

state |#

 with probability p

, we shall think of R

as simply a pre-probability

(Sec. 15.2). Assume that while the particle is inside the interferometer, during the

time interval from t

to t

, the interaction with the environment gives rise to unitary

transformations

|c|#

%→|c



|ζ

, |d|#

%→|d



|η

, (26.18)

26.4 Random environment 355

where we are again assuming, as in (26.4) and (26.14), that the environment has

a negligible inﬂuence on the particle wave packets |c



 and |d



. Because the time

evolution is unitary, {|ζ

} and {|η

} are orthonormal bases of E.

Let the state of the particle and the environment at t

be given by a density matrix

= [¯a] ⊗ R

, (26.19)

where |¯a=(|c+|d)/

√

2 is the state of the particle when it emergesfrom the ﬁrst

beam splitter. At t

, just before the particle leaves the interferometer, the density

matrix resulting from unitary time evolution of the total system will be







c



|⊗|ζ

ζ

|+|d



d



|⊗|η

η

+|c



d



|⊗|ζ

η

|+|d



c



|⊗|η

ζ



. (26.20)

Taking a partial trace gives the expression

= Tr

) =





c



|+|d



d



|+α|c



d



|+α

∗



c





, (26.21)

for the reduced density matrix of the particle at time t

, where

α =



η

|ζ

. (26.22)

The expression (26.21) is formally identical to (26.8), but the complex parameter

α is now a weighted average of a collection of complex numbers, the inner products

{η

|ζ

}, each with magnitude less than or equal to 1. Consider the case in which

η

|ζ

=e

iφ

, (26.23)

that is, the interaction with the environment results in nothing but a phase differ-

ence between the wave packets of the particle in the c and d arms. Even though

|η

|ζ

| = 1 for every j, the sum (26.22) will in general result in |α| < 1, and if

the sum includes a large number of random phases, |α| can be quite small. Hence

a random environment can produce decoherence even in circumstances in which a

nonrandom environment (as discussed in Secs. 26.2 and 26.3) does not.

The basis {|#

} in which R

is diagonal is useful for calculations, but does not

actually enter into the ﬁnal result for ρ

. To see this, rewrite (26.18) in the form

|c⊗|#%→|c



⊗U

|#, |d⊗|#%→|d



⊗U

|#, (26.24)

where |# is any state of the environment, and U

and U

are unitary transforma-

tions on E. Then (26.22) can be written in the form

α = Tr



†



, (26.25)

which makes no reference to the basis {|#

}.

356 Decoherence and the classical limit

26.5 Consistency of histories

Consider a family of histories at times t

< t

with support

= [ψ

] ( [c] ( [c



] ( [e],

= [ψ

] ( [d] ([d



] ( [e],

= [ψ

] ( [c] ( [c



] ( [ f ],

= [ψ

] ( [d] ([d



] ( [ f ],

(26.26)

where [ψ

] is the initial state |a|# in (26.6), and the unitary dynamics is that of

Sec. 26.2. The chain operators for histories which end in [e] are automatically

orthogonal to those of histories which end in [ f ]. However, when the ﬁnal states

are the same, the inner products are

K(Y

), K(Y

)=α/4 =−K(Y

), K(Y

), (26.27)

where α is the parameter deﬁned in (26.5), which appears in the density matrix

(26.8) or (26.9). Equation (26.27) is also valid in the case of multiple interactions

with the environment, where α is given by (26.15). And it holds for the random

environment discussed in Sec. 26.4, with α deﬁned in (26.22), provided one re-

deﬁnes the histories in (26.26) by eliminating the initial state [ψ

], so that each

history begins with [c]or[d]att

, and uses the density matrix "

, (26.19), as an

initial state at time t

in the consistency condition (15.48). In this case the operator

inner product used in (26.27) is , 

, as it involves the density matrix "

,see

(15.48).

If there is no interaction with the environment, then α = 1 and (26.27) implies

that the family (26.26) is not consistent. However, if α is very small, even though

it is not exactly zero, one can say that the family (26.26) is approximately consis-

tent, or consistent for all practical purposes, for the reasons indicated at the end

of Sec. 10.2: one expects that by altering the projectors by small amounts one can

produce a nearby family which is exactly consistent, and which has essentially the

same physical signiﬁcance as the original family.

This shows that the presence of decoherence may make it possible to discuss

the time dependence of a quantum system using a family of histories which in the

absence of decoherence would violate the consistency condtions and thus not make

sense. This is an important consideration when one wants to understand how the

classical behavior of macroscopic objects is consistent with quantum mechanics,

which is the topic of the next section.

26.6 Decoherence and classical physics

The simple example discussed in the preceding sections of this chapter illustrates

two important consequences of decoherence: it can destroy interference effects,

26.6 Decoherence and classical physics 357

and it can render certain families of histories of a subsystem consistent, or at least

approximately consistent, when in the absence of decoherence such a family is in-

consistent. There is an additional important effect which is not part of decoherence

as such, for it can arise in either a classical or a quantum system interacting with its

environment: the environment perturbs the motion of the system one is interested

in, typically in a random way. (A classical example is Brownian motion, Sec. 8.1.)

The laws of classical mechanics are simple, have an elegant mathematical form,

and are quite unlike the laws of quantum mechanics. Nonetheless, physicists be-

lieve that classical laws are only an approximation to the more fundamental quan-

tum laws, and that quantum mechanics determines the motion of macroscopic

objects made up of many atoms in the same way as it determines the motion of

the atoms themselves, and that of the elementary particles of which the atoms are

composed. However, showing that classical physics is a limiting case of quantum

physics is a nontrivial task which, despite considerable progress, is not yet com-

plete, and a detailed discussion lies outside the scope of this book. The following

remarks are intended to give a very rough and qualitative picture of how the cor-

respondence between classical and quantum physics comes about. More detailed

treatments will be found in the references listed in the bibliography.

A macroscopic object such as a baseball, or even a grain of dust, is made up of

an enormous number of atoms. The description of its motion provided by classi-

cal physics ignores most of the mechanical degrees of freedom, and focuses on a

rather small number of collective coordinates. These are, for example, the center

of mass and the Euler angles for a rigid body, to which may be added the vibra-

tional modes for a ﬂexible object. For a ﬂuid, the collective coordinates are the

hydrodynamic variables of mass and momentum density, thought of as obtained by

“coarse graining” an atomic description by averaging over small volumes which

still contain a very large number of atoms. It is important to note that the classi-

cal description employs a very special set of quantities, rather than using all the

mechanical degrees of freedom.

It is plausible that properties represented by classical collective coordinates, such

as “the mass density in region X has the value Y”, correspond to projectors onto

subspaces of a suitable Hilbert space. These subspaces will have a very large di-

mension, because the classical description is relatively coarse, and there will not

be a unique projector corresponding to a classical property, but instead a collection

of projectors (or subspaces), all of which correspond within some approximation

to the same classical property.

In the same way, a classical property which changes as a function of time will

be associated with different projectors as time progresses, and thus with a quantum

history. The continuous time variable of a classical description can be related to the

discrete times of a quantum history in much the same way as a continuous classical

358 Decoherence and the classical limit

mass distribution is related to the discrete atoms of a quantum description. Just

as a given classical property will not correspond to a unique quantum projector,

there will be many quantum histories, and families of histories, which correspond

to a given classical description of the motion, and represent it to a fairly good

approximation. The term “quasi-classical” is used for such a quantum family and

the histories which it contains.

In order for a quasi-classical family to qualify as a genuine quantum description,

it must satisfy the consistency conditions. Can one be sure that this is the case?

Gell-Mann, Hartle, Brun, and Omn

es (see references in the bibliography) have

studied this problem, and concluded that there are some fairly general conditions

under which one can expect consistency conditions to be at least approximately

satisﬁed for quasi-classical families of the sort one encounters in hydrodynamics

or in the motion of rigid objects. That such quasi-classical families will turn out

to be consistent is made plausible by the following consideration. Any system of

macroscopic size is constantly in contact with an environment. Even a dust par-

ticle deep in interstellar space is bombarded by the cosmic background radiation,

and will occasionally collide with atoms or molecules. In addition to an external

environment of this sort, macroscopic systems have an internal environment con-

stituted by the degrees of freedom left over when the collective coordinates have

been speciﬁed. Both the external and the internal environment can contribute to

processes of decoherence, and these can make it very hard to observe quantum in-

terference effects. While the absence of interference, which is signaled by the fact

that the density matrix of the subsystem is (almost) diagonal in a suitable repre-

sentation, is not the same thing as the consistency of a suitable family of histories,

nonetheless the two are related, as suggested by the example considered earlier

in this chapter, where the same parameter α characterizes both the degree of co-

herence of the particle when it leaves the interferometer, and also the extent to

which certain consistency conditions are not fulﬁlled. The effectiveness of this

kind of decoherence is what makes it very difﬁcult to design experiments in which

macroscopic objects, even those no bigger than large molecules, exhibit quantum

interference.

If a quasi-classical family can be shown to be consistent, will the histories in it

obey, at least approximately, classical equations of motion? Again, this is a non-

trivial question, and we refer the reader to the references in the bibliography for

various studies. For example, Omn

es has published a fairly general argument that

classical and quantum mechanics give similar results if the quantum projectors cor-

respond (approximately) to a cell in the classical phase space which is not too small

and has a fairly regular shape, provided that during the time interval of interest the

classical equations of motion do not result in too great a distortion of this cell. This

last condition can break down rather quickly in the presence of chaos, a situation in

26.6 Decoherence and classical physics 359

which the motion predicted by the classical equations depends in a very sensitive

way upon initial conditions.

Classical equations of motion are deterministic, whereas a quantum description

employing histories is stochastic. How can these be reconciled? The answer is that

the classical equations are idealizations which in appropriate circumstances work

rather well. However, one must expect the motion of any real macroscopic system

to show some effects of a random environment. The deterministic equations one

usually writes down for classical collective coordinates ignore these environmental

effects. The equations can be modiﬁed to allow for the effects of the environment

by including stochastic noise, but then they are no longer deterministic, and this

narrows the gap between classical and quantum descriptions. It is also worth keep-

ing in mind that under appropriate conditions the quantum probability associated

with a suitable quasi-classical history of macroscopic events can be very close to

1. These considerations would seem to remove any conﬂict between classical and

quantum physics with respect to determinism, especially when one realizes that the

classical description must in any case be an approximation to some more accurate

quantum description.

In conclusion, even though many details have not been worked out and much

remains to be done, there is no reason at present to doubt that the equations of

classical mechanics represent an appropriate limit of a more fundamental quantum

description based upon a suitable set of consistent histories. Only certain aspects

of the motion of macroscopic physical bodies, namely those described by appro-

priate collective coordinates, are governed by classical laws. These laws provide

an approximate description which, while quite adequate for many purposes, will

need to be supplemented in some circumstances by adding a certain amount of

environmental or quantum noise.

Quantum theory and reality

27.1 Introduction

The connection between human knowledge and the real world to which it is (hope-

fully) related is a difﬁcult problem in philosophy. The purpose of this chapter is

not to discuss the general problem, but only some aspects of it to which quantum

theory might make a signiﬁcant contribution. In particular, we want to discuss the

question as to how quantum mechanics requires us to revise pre-quantum ideas

about the nature of physical reality. This is still a very large topic, and space will

permit no more than a brief discussion of some of the signiﬁcant issues.

Physical theories should not be confused with physical reality. The former are,

at best, some sort of abstract or symbolic representation of the latter, and this is

as true of classical physics as of quantum physics. The phase space used to repre-

sent a classical system and the Hilbert space used for a quantum system are both

mathematical constructs, not physical objects. Neither planets nor electrons inte-

grate differential equations in order to decide where to go next. Wave functions

exist in the theorist’s notebook and not, unless in some metaphorical sense, in the

experimentalist’s laboratory. One might think of a physical theory as analogous

to a photograph, in that it contains a representation of some object, but is not the

object itself. Or one can liken it to a map of a city, which symbolizes the locations

of streets and buildings, even though it is only made of paper and ink.

We can comprehend (to some extent) with our minds the mathematical and log-

ical structure of a physical theory. If the theory is well developed, there will be

clear relationships among the mathematical and logical elements, and one can dis-

cuss whether the theory is coherent, logical, beautiful, etc. The question of whether

a theory is true, its relationship to the real world “out there”, is more subtle. Even if

a theory has been well conﬁrmed by experimental tests, as in the case of quantum

mechanics, believing that it is (in some sense) a true description of the real world

requires a certain amount of faith. A decision to accept a theory as an adequate,

360

27.2 Quantum vs. classical reality 361

or even as an approximate representation of the world is a matter of judgment

which must inevitably move beyond issues of mathematical proof, logical rigor,

and agreement with experiment.

If a theory makes a certain amount of sense and gives predictions which agree

reasonably well with experimental or observational results, scientists are inclined

to believe that its logical and mathematical structure reﬂects the structure of the

real world in some way, even if philosophers will remain permanently sceptical.

Granted that all theories are eventually shown to have limitations, we nonethe-

less think that Newton’s mechanics is a great improvement over that of Aristotle,

because it is a much better reﬂection of what the real world is like, and that relativ-

ity theory improves upon the science of Newton because space-time actually does

have a structure in which light moves at the same speed in any inertial coordinate

system. Theories such as classical mechanics and classical electromagnetism do a

remarkably good job within their domains of applicability. How can this be under-

stood if not by supposing that they reﬂect something of the real world in which we

live?

The same remarks apply to quantum mechanics. Since it has a consistent math-

ematical and logical structure, and is in good agreement with a vast amount of

observational and experimental data, it is plausible that quantum theory is a better

reﬂection of what the real world is like than the classical theories which preceded

it, and which could not explain many of the microscopic phenomena that are now

understood using quantum methods. The faith of the physicist is that the real world

is something like our best theories, and at the present time it is universally agreed

that quantum mechanics is a very good theory of the physical world, better than

any other currently available to us.

27.2 Quantum vs. classical reality

What are the main respects in which quantum mechanics differs from classical

mechanics? To begin with, quantum theory employs wave functions belonging to

a Hilbert space, rather than points in a classical phase space, in order to describe

a physical system. Thus a quantum particle, in contrast to a classical particle,

Secs. 2.3 and 2.4, does not possess a precise position or a precise momentum.

In addition, the precision with which either of these quantities can be deﬁned is

limited by the Heisenberg uncertainty principle, (2.22). This does not mean that

quantum entities are “fuzzy” and ill-deﬁned, for a ray in the Hilbert space is as

precise a speciﬁcation as a point in phase space. What it does mean is that the clas-

sical concepts of position and momentum can only be used in an approximate way

when applied to the quantum domain. As pointed out in Sec. 2.4, the uncertainty

principle refers primarily to the fact that quantum entities are described by a very

362 Quantum theory and reality

different mathematical structure than are classical particles, and only secondarily

to issues associated with measurements. The limitations on measurements come

about because of the nature of quantum reality, and the fact that what does not exist

cannot be measured.

A second respect in which quantum mechanics is fundamentally different from

classical mechanics is that the basic classical dynamical laws are deterministic,

whereas quantum dynamical laws are, in general, stochastic or probabilistic, so

that the future behavior of a quantum system cannot be predicted with certainty,

even when given a precise initial state. It is important to note that in quantum

theory this unpredictability in a system’s time development is an intrinsic feature

of the world, in contrast to examples of stochastic time development in classical

physics, such as the diffusion of a Brownian particle (Sec. 8.1). Classical unpre-

dictability arises because one is using a coarse-grained description where some

information about the underlying deterministic system has been thrown away, and

there is always the possibility, in principle, of a more precise description in which

the probabilistic element is absent, or at least the uncertainties reduced to any ex-

tent one desires. By contrast, the Born rule or its extension to more complicated

situations, Chs. 9 and 10, enters quantum theory as an axiom, and does not result

from coarse graining a more precise description. To be sure, there have been ef-

forts to replace the stochastic structure of quantum theory with something more

akin to the determinism of classical physics, by supplementing the Hilbert space

with hidden variables. But these have not turned out to be very fruitful, and, as dis-

cussed in Ch. 24, the Bell inequalities indicate that such theories can only restore

determinism at the price of introducing nonlocal inﬂuences violating the principles

of special relativity.

Of course, there is no reason to suppose that quantum mechanics as understood

at the present time is the ultimate theory of how the world works. It could be that at

some future date its probabilistic laws will be derived from a superior theory which

returns to some form of determinism, but it is equally possible that future theories

will continue to incorporate probabilistic time development as a fundamental fea-

ture. The fact that it was only with great reluctance that physicists abandoned

classical determinism in the course of developing a theory capable of explaining

experimental results in atomic physics strongly suggests, though it does not prove,

that stochastic time development is part of physical reality.

27.3 Multiple incompatible descriptions

The feature of quantum theory which differs most from classical physics is that it

allows one to describe a physical system in many different ways which are incom-

patible with one another. Under appropriate circumstances two (or more) incom-

27.3 Multiple incompatible descriptions 363

patible descriptions can be said to be true in the sense that they can be derived in

different incompatible frameworks starting from the same information about the

system (the same initial data), but they cannot be combined in a single description,

see Sec. 16.4. There is no really good classical analog of this sort of incompatibil-

ity, which is very different from what we ﬁnd in the world of everyday experience,

and it suggests that reality is in this respect very different from anything dreamed

of prior to the advent of quantum mechanics.

As a speciﬁc example, consider the situation discussed in Sec. 18.4 using

Fig. 18.4, where a nondestructive measurement of S

is carried out on a spin-half

particle by one measuring device, and this is followed by a later measurement of

using a second device. There is a framework F, (18.31), in which it is possi-

ble to infer that at the time t

when the particle was between the two measuring

devices it had the property S

=+1/2, and another, incompatible framework G,

(18.33), in which one can infer the property S

=+1/2att

. But there is no way

in which these inferences, even though each is valid in its own framework, can

be combined, for in the Hilbert space of a spin-half particle there is no subspace

which corresponds to S

=+1/2ANDS

=+1/2, see Sec. 4.6. Thus we have

two descriptions of the same quantum system which because of the mathematical

structure of quantum theory cannot be combined into a single description.

It is not the multiplicity of descriptions which distinguishes quantum from clas-

sical mechanics, for multiple descriptions of the same object occur all the time in

classical physics and in everyday life. A teacup has a different appearance when

viewed from the top or from the side, and the side view depends on where the han-

dle is located, but there is never any problem in supposing that these different de-

scriptions refer to the same object. Or consider a macroscopic body which is spin-

ning. One description might specify the z-component L

of its angular momentum,

and another the x-component L

. In classical physics, two correct descriptions of

a single object can always be combined to produce a single, more precise descrip-

tion, and if this process is continued using all possible descriptions, the result will

be a unique exhaustive description which contains each and every detail of every

true description. In the case of a mechanical system at a single time, the unique

exhaustive description corresponds to a single point in the classical phase space.

Any true description can be obtained from the unique exhaustive description by

coarsening it, that is, by omitting some of the details. Thus specifying a region

in the phase space rather than a single point produces a coarser description of a

mechanical system.

For the purposes of the following discussion it is convenient to refer to the idea

that there exists a unique exhaustive description as the principle of unicity, or sim-

ply unicity. This principle implies that every conceivable property of a particular

physical system will be either true or false, since it either is or is not contained in,