Mathematical Foundations of Quantum Mechanics 1812.4 Just Few Words about the Algebraic Approach
The fundamental theorem 2.3.6.6 of Stone-von Neumann and Mackey is stated in
the jargon of theoretical physics as follows:
“all irreducible representations of the CCRs with a finite, and fixed, number of
degrees of freedom are unitarily equivalent”.
The expressionunitarily equivalentrefers to the existence of the Hilbert-space
isomorphismU, and the finite number of degrees of freedom is the dimension of
the Lie algebra spanned by the generatorsI,Xk,Pk.
What happens then in infinite dimensions? This is the case when dealing with
quantum fields,wherethe2n+1 generatorsI,Xk,Pk(k=1, 2 ,...,n), are replaced
by acontinuumof generators, the so-calledfield operators at fixed timeand the
conjugated momentum at fixed time: I,Φ(f),Π(g) which are smeared by
arbitrary functionsf,g∈C 0 ∞(R^3 ). HereR^3 is the rest space of a given reference
frame in the spacetime. Those field operators satisfy commutation relations similar
to the ones ofXkandPk(e.g., see[26; 27; 28]). Then the Stone–von Neumann
theorem no longer holds. In this case, theoretical physicists would say that
“there exist irreducible non-equivalent CCR representations with an infinite
number of degrees of freedom”.
What happens in this situation, in practice, is that one finds twoisomorphic
∗-algebras of field operators, the one generated by Φ(f),Π(g)intheHilbertspace
Hand the other generated by Φ′(f),Π′(g) in the Hilbert spaceH′that admitno
Hilbert spaceisomorphismU:H′→Hsatisfying:
UΦ′(f)U−^1 =Φ(f),UΠ′(g)U−^1 =Π(g) for any pairf,g∈C 0 ∞(R^3 ).Pairs of this kind are called(unitarily) non-equivalent. Jumping from the finite-
dimensional case to the infinite-dimensional one corresponds to passing from Quan-
tum Mechanics to Quantum Field Theory (possibly relativistic, and on curved
spacetime[ 28 ]). The presence of non-equivalent representations of one single phys-
ical system shows that a formulation in a fixed Hilbert space is fully inadequate,
at least because it insists on a fixed Hilbert space, whereas the physical system is
characterized by a more abstract object: An algebra of observables which may be
represented in different Hilbert spaces in terms of operators. These representations
are not unitarily equivalent and none can be considered more fundamental than
the remaining ones. We must abandon the structure of Hilbert space in order to
lay the foundations of quantum theories in broader generality.
This program has been widely developed (see e.g.,[13; 25; 26; 27]), starting
from the pioneering work of von Neumann himself, and is nowadays calledalgebraic
formulation of quantum (field) theories. Within this framework it was possible to