------------------------------------------- Inequivalent representations of the CCR/CAR ------------------------------------------- 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.