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Inequivalent representations of the CCR/CAR
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Ordinary quantum mechanics of a particle with N flavors can be written
in terms of creation and annihilation operators for the 3N modes of an
associated reference harmonic oscillator. The field case, on the other
hand, is characterized by the fact that there are infinitely many modes.
If the creation and annihilation operators are those in the action
or Hamiltonian defining the QFT, the different modes are traditionally
referred to as 'bare particles', though this is not recommended for
reasons discussed elsewhere in this FAQ. If the creation and
annihilation operators are properly renormalized so that they
create and annihilate physical particles from the physical vacuum,
the modes are referred to as 'dressed particles'; only these have
physical relevance.
A state in which k modes are excited is called a k-particle state.
In many states of interest, however, (the most prominent ones being
the coherent states) infinitely many modes are excited (although the
notion of infinitely particles is strained in this case). Thus one
needs to cater in the formalism for states with arbitrarily many or
even infinitely many modes. This has subtle consequences, which
account for the big difference between quantum field theory and
ordinary quantum mechanics.
The canonical commutation rules (CCR) for creation and annihilation
operators in field theory take in the simplest case (countably many
modes, corresponding to fields confined to a bounded region) the form
[a(k),a^*(l)] = delta_kl, k,l=0,1,2,... (1)
The Stone-von Neumann theorem, which guarantees that the canonical
commutation relations of quantum mechanics (or their unitary version,
the Weyl relations) have a unique unitary representation apart from
unitary transformations, fails for systems of infinitely many degrees
of freedom.
The reason for this is that the natural representation space for
creation and annihilation operators is the vector space consisting
of all formal linear combinations
sum psi(n1,n2,n3,...) |n1,n2,n3,...>
with _arbitrary_ complex coefficients psi(n1,n2,n3,...), on which
a(k) and a^*(l) act as
a(k)|n1,....,n_k,...> = sqrt(n_k)|n1,....,n_k - 1,...>,
a*(l)|n1,....,n_l,...> = sqrt(1+n_l)|n1,....,1+n_l,...>.
This vector space V has no natural Hilbert space structure.
To provide a definite inner product, one must select a suitable
subspace where this inner product can be defined.
This allows many choices; the choice usually discussed in QFT treatises
is Fock space, where only basis vectors |n1,....,n_k,0,0,...>
with finitely many particles are allowed, and these basis vectors are
declared orthonormal. As a result, Fock space contains only
the linear combinations
sum psi(n1,n2,n3,...,n_k) |n1,n2,n3,...,n_k>
where k is variable and
sum |psi(n1,n2,n3,...,n_k)|^2 is finite.
Unfortunately, if this choice is made for the representation of the
bare creation and annihilation operators, it excludes the states
relevant for the physical, interacting situation. This is the
essential message of Haag's no interaction theorem.
R. Haag
On quantum field theories
Mat.-Fys. Medd. Kong. Danske Videns. Selskab 29 (12) (1955), 1-37.
http://cdsweb.cern.ch/record/212242/files/p1.pdf
Indeed, the physical states lie in a different, inequivalent unitary
representation, characterized by a different subspace of V. This
subspace is generated by applying to the physical (= renormalized)
vacuum state the dressed (= renormalized) creation operators
an arbitrary number of times, then taking all finite linear
combinations, and finally taking the closure with respect to the
innner product in which all a^*(n_1)...a^*(n_k)|vac> are orthonormal.
In general, this Hilbert space has only the null vector (_not_ the
vacuum) in common with the Fock space, even for the simplest
(i.e.,quadratic) Hamiltonians and actions. This case is well understood,
giving rise to the theory of quasiparticles and in particular of
superconductivity. For example (counting modes by signed nonzero
integers for simplicity - they become momenta in the infinite volume
limit), if the bare a(k) and b(k) satisfy CCR then do the dressed
annihilation operators
alp(k) = A(k) a(k) - B(-k) b*(-k),
bet(k) = A(k) b(k) - B(-k) a*(-k),
and their formal adjoints
alp^*(k) = A(-k) a^*(k) - B(k) b(-k),
bet^*(k) = A(-k) b^*(k) - B(k) a(-k),
provided that A(k), B(k) are real numbers satisfying
A(k)^2 - B(k)^2 = 1,
or, equivalently, that
A(k) = cosh(theta(k)), B = sinh(theta(k)).
If there were only finitely many modes, we could define
in Fock space the unitary operator
G = exp [- sum_k theta(k) (a(k)b(-k) - b*(-k)a*(k))],
and verify that
alp(k) = G a(k) G^{-1},
bet(k) = G b(k) G^{-1},
showing that we get an equivalent representation of the CCR.
We could deduce that
|vac> := G|>,
where |> is the bare vacuum, is the dressed vacuum on which
alp and bet act naturally. The dressed states were simply be
the images of the bare states under the Bogoliubov operator G.
Unfortunately, if there are infinitely many modes, G can no
longer be consistently defined as an operator in Fock space,
and the infinite-dimensional version of this scenario breaks
down. Ignoring this, one would find all sorts of infinities.
Mathematically, however, one simply changed the unitary
representation - G does not exist although the dressed
representation exists.
Physicists say that the above computations hold 'formally',
and mean (if a mathematician tries to give it a precise meaning)
that it holds in finite mode approximations but does not survive
the limit although they usually formulate it in the meaningless,
limit form.
The canonical anticommutation rules (CAR) also have the form (1),
except that the commutator is replaced by an anticommutator.
All statements above are valid with appropriate modifications;
the most important one being that occupation numbers are now
restricted to 0 and 1, and the definition of a^*(l) has 1-n_l in
place of 1+n_l.
For more details see the book
H. Umezawa, H. Matsumoto, and M. Tachiki,
Thermo Field Dynamics and Condensed States,
North Holland 1982.