From Classical Mechanics to Quantum Field Theory
148 From Classical Mechanics to Quantum Field Theory. A Tutorial At this juncture we are ready to define von Neumann algebras. D ...
Mathematical Foundations of Quantum Mechanics 149 2.3.5.2 Lattices of von Neumann algebras To conclude this elementary mathemati ...
150 From Classical Mechanics to Quantum Field Theory. A Tutorial indicated byR, called thevon Neumann algebra of observables(tho ...
Mathematical Foundations of Quantum Mechanics 151 Remark 2.3.37.It is possible to prove that a von Neumann algebra is always a d ...
152 From Classical Mechanics to Quantum Field Theory. A Tutorial (2)If adding the spin space (for instance dealing with an elect ...
Mathematical Foundations of Quantum Mechanics 153 (a)Hadmits the following direct decomposition into closed pairwise orthogo- na ...
154 From Classical Mechanics to Quantum Field Theory. A Tutorial withR, so it must belong to the centre for (SS2) and thus it be ...
Mathematical Foundations of Quantum Mechanics 155 Remark 2.3.42. (a)A fundamental requirement is that the superselection charges ...
156 From Classical Mechanics to Quantum Field Theory. A Tutorial If there is a superselection structure, we have the decompositi ...
Mathematical Foundations of Quantum Mechanics 157 Proof. (a)is obvious from Proposition 2.3.23, as restricting a stateρonL(H)to ...
158 From Classical Mechanics to Quantum Field Theory. A Tutorial the standard procedure (which is nothing but the trace procedur ...
Mathematical Foundations of Quantum Mechanics 159 2.3.6.1 Wigner and Kadison theorems, groups of symmetries Consider a quantum s ...
160 From Classical Mechanics to Quantum Field Theory. A Tutorial (a)For every Wigner symmetrysW there is an operatorU:H→H,which ...
Mathematical Foundations of Quantum Mechanics 161 The meaning ofs∗(A) should be evident: the probability that the observable s∗( ...
162 From Classical Mechanics to Quantum Field Theory. A Tutorial TheactionofthissymmetryontheobservableXkturns out to be su∗(Xk) ...
Mathematical Foundations of Quantum Mechanics 163 That is the same as requiring that there are numbersχh∈U(1),ifh∈G, such that ω ...
164 From Classical Mechanics to Quantum Field Theory. A Tutorial unitary projective representation ofGin a complex Hilbert space ...
Mathematical Foundations of Quantum Mechanics 165 (a)γis equivalent to a strongly continuous unitary representationRr→Vr of th ...
166 From Classical Mechanics to Quantum Field Theory. A Tutorial technical results of very different nature which are very usefu ...
Mathematical Foundations of Quantum Mechanics 167 One parameter unitary group generated by selfadjoint operators can be used to ...
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