Yes, in QM/QFT we always have some group G that is required to have representati
ID: 1382370 • Letter: Y
Question
Yes, in QM/QFT we always have some group G that is required to have representation on a Hilbert space. This group in particular usually includes time translation and space translation operators as its generators which are interpreted as Hamiltonian and Momentum observables respectively. In fact all observables of the theory come from the generators of the group G.
In a classical field theory phase space is a space of fields. Observables are functions on this space, and hence are functionals of fields. A subalgebra of the Poisson algebra of these observables generates group G (modulo some discrete symmetries). So upon quantization, G gets represented on space of states through a representation of the (Poisson) algebra of fields. It is this thing which distinguishes a QFT from abstract representation theory of group G. We not only require that space of states H be a representation space of G but also that its action on H be written in terms of action of some algebra of quantum fields on a given space time.
Two representations of algebra of quantum fields may not be equivalent to each other. Uniqueness of representation is not a problem with finitely many degrees of freedom. When number of degrees of freedom are infinite (which is usually the case in a typical QFT) then one usually works in Fock space representation.
In physics one usually looks for projective unitary representation of G which are not true representations but representations modulo phase or equivalently unitary representation of some central extension of G. In some cases for the consistent definition of the theory it may be required that the representation of G on space of states be a true representation.
Explanation / Answer
Representation theory is essential to define the relevant physical observables of a theory. The reason is that all separable Hilbert spaces of the same dimension are isomorphic (hence the same apart from labeling). Thus one needs extra structure to distinguish the interesting physics.
This is done by specifying a ''dynamical'' Lie algebra of important observables, whose physical interpretation is given by the form of the Lie bracket.
For example, the canonical commutation rules define the formal meaning of position and momentum in QM, and of free fields in QFT. The Lie algebra so(3) defines angular momentum, the Poincare group defines 4-momentum and 4-angular momentum, su(2) defines isospin, su(3) defines color, etc.
The particular unitary (projective) representation on a Hilbert space then determines the quantum numbers and particle contents of a theory.
The Lie algebra of the symmetry group is usually much smaller than the dynamical Lie algebra, but plays another important role.
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