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Indistinguishable particles and fields
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Commonly, outside of quantum field theory, there is a lot of
imprecision in talking about ''indistinguishable'' (or ''identical'')
particles, even in serious work.
Historically, indistinguishable particles were introduced in order to
explain the failure of the thermodynamics of a Newtonian N-particle
system to describe the absence of an entropy increase when mixing two
volumes of the same substance. This assumption, which has no logical
basis in classical mechanics but appears as an ad hoc device to save the
theory, drastically changes the state space of the system (classically,
the phase space, quantum mechanically, the Hilbert space).
The first quantum manifestation of indistinguishable particles was the
Pauli exclusion principle, which st ates that wave functions of a
nonrelativistic N-electron system must be completely
antisymmetrized. The same holds for other kinds of indistinguishable
fermions, while for a system of indistinguishable bosons, the wave
functions must be completely symmetrized.
As a result, in nonrelativistic quantum mechanics, the state space of
indistinguishable particles is very different from that of
distinguishable particles; nothing particle-like survives this change of
basic setting.
The change completely destroys the very basis of a reasonable particle
concept: the possibility to assign observables (self-adjoint operators)
to the position of a particle inside a multiparticle system. There are
_no_ such observables. The reason is that observables must map correctly
(anti)symmetrized multiparticle wave functions again to correctly
(anti)symmetrized multiparticle wave functions.
For example, multiplication by x_i, the standard position observable for
particle i in the distinguishable case, does not have this property. And
if one correctly (anti)symmetrizes the result, the resulting operator
only describes the center of mass of the system, not a property of
individual particles. Similarly, one can consider multiplication by
f(x_i) for some arbitrary function f of a single position, and
(anti)symmetrize the result. As result one gets the value of f(x),
averaged over all positions x. In other words, the observables that one
still can form, correspond to integrals of the form
N(f)= integral dx f(x)Rho(x),
where Rho(x) denotes the particle density at the position x. The alleged
particle theory has no particle-like observable properties at all -
these dissolved under our analysis and lead instead to a particle
density field. Thus we are forced to acknowledge that a quantum theory
of indistinguishable ''particles'' is in fact a quantum field theory,
and _only_ a quantum field theory!
Indeed, this is made formally precise by the concept of ''second
quantization'', usually introduced in statistical mechanics, where the
field view is indispensable. In the nonrelativistic case of a field of
structureless and spinless particles, it is easily checked that the
particle density can be written as Rho(x)=a^*(x)a(x), where a(x) is the
annihilator operator (or rather operator-valued distribution) in the
position representation.
On the other hand, if one starts with an annihilator field a(x)
satisfying canonical commutation rules, one gets indistinguishable
particles for free. Thus indistinguishable particles are one of the most
elementary consequences of the quantum field concept!
Nothing essential changes in the above discussion when one considers
relativistic quantum mechanics of massive particles in the (then
existing) position representation.
Thus the concept of indistinguishable particles is completely superseded
by the concept of a quantum field. The latter gives the right intuition
about the meaning of the formalism, and the former (which is difficult
to justify and even more difficult to interpret intuitively) is
completely dispensable.
How then do the familiar ideas about particles reappear in the quantum
field setting? This is succinctly answered by quoting Steven Weinberg:
''In its mature form, the idea of quantum field theory is that quantum
fields are the basic ingredients of the universe, and particles are just
bundles of energy and momentum of the fields.'' (see p.2 of his essay,
''What is Quantum Field Theory, and What Did We Think It Is?''
http://arxiv.org/pdf/hep-th/9702027v1)
We see individual photons when they directly fall into our eyes. But
these photons are not described by single photon states of QED but are
effective photons (part of a very complex state of the electromagnetic
field) distinguished by their position. Particles we can see through an
electron microscope, say, appear there as lumps with slightly fuzzy
boundaries, described by a field density. In the most precise
experiments, we hear sounds, read pointers or digital numbers, or see
ionization tracks that are only indirectly related to pointlike
particles - whose existence we pretend to infer, but in fact simply (and
falsely) assume.