By Harald Stumpf, Thomas Borne (auth.)
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Extra resources for Composite Particle Dynamics in Quantum Field Theory
IpJIpk+1 ... IPnB(t - tpk+l ) k=l n-l - L(-1)kIp1 ... IpJIpk+I···IpnB(t-tPk) k=l +(-ltIpI" . IPnJ8(tPn - t)] = L sgn(p )8( tpI ... )] Ip2 ... ,IPn pESn +(-1)nIpI" . IPn_1 [IPnJ8(tpn - t) - JIpnB(t - tpn )] n-l + L(-1)k+ 1 Ip1 .. , Ipk _l [Ipk,J]+Ipk+1 .. 68) it follows OtT(JIl ... In) = T(jh ... 69) n = - L sgn(p)8( tpI ... , t pn ) L( _l)k IpI ... Ipk _l [Ipk , J]+Ipk+1 ... IPn 6( t - tpk ) pESn k=l T(jh ... I n ) n - L sgn(p)8(t P1 pESn ••• , I t pn ) L( _l)k IpI ... Pk ... IpnAlpk J6(t - tpk ) k=l If one transforms the sequence of indices (1,2, ...
However, these scalar products in functional space are completely independent from those of the state space IHl and cannot be used for the construction of an isometric map between IHl and OCF . The functional space is an artificial construction and one cannot expect to guess the physical scalar products by a priori constructions in functional space, as these physical scalar products depend on the system dynamics itself. The theory of isometric maps will be developed in Chapter 3. 17) a map between the physical state space IHl and the functional state space OCF is established.
Bognar [Bogn 74]. Hence all the physics and "nonphysics" can be covered by this construction and we can base our probability interpretation on it. As the physical subspace llifph of \HI we define the closed linear hull of all eigenstates with positive norm in \HI with respect to a suitable topology, see Bognar [Bogn 74]. For simplicity we consider only the Hamiltonian H.
Composite Particle Dynamics in Quantum Field Theory by Harald Stumpf, Thomas Borne (auth.)