Wavefunction Collapse and Conservation Laws

It is emphasized that the collapse postulate of standard quantum theory can violate conservation of energy-momentum and there is no indication from where the energy-momentum comes or to where it goes. Likewise, in the Continuous Spontaneous Localization (CSL) dynamical collapse model, particles gain energy on average. In CSL, the usual Schrödinger dynamics is altered so that a randomly fluctuating classical field interacts with quantized particles to cause wavefunction collapse. In this paper it is shown how to define energy for the classical field so that the average value of the energy of the field plus the quantum system is conserved for the ensemble of collapsing wavefunctions. While conservation of just the first moment of energy is, of course, much less than complete conservation of energy, this does support the idea that the field could provide the conservation law balance when events occur. 1 The Collapse Postulate and Nonconservation of Energy-Momentum It is not generally appreciated that the collapse postulate of standard quantum theory (SQT) can violate the geometric conservation laws, e.g., the conservation of energy and momentum[1]. In quantum theory, conservation of a physical quantity represented by the operator Q is the statement that the probability distribution of Q’s eigenvalues q, |〈q|ψ, t〉|, is constant in time. For the Schrödinger evolution of a statevector, conservation is guaranteed by the vanishing of the commutator of Q with the Hamiltonian. This ensures that 〈ψ, t|F (Q)|ψ, t〉 = 〈ψ, 0|F (Q)|ψ, 0〉, where F is an arbitrary function of Q: the constancy of |〈q|ψ, t〉| immediately follows. (It will be useful for later purposes to note here that this is also equivalent to constancy of the expectation value of arbitrary integer powers of Q, which in turn is equivalent to constancy of the expectation value of the generating function G ≡ exp iaQ.) However, according to SQT, the Schrödinger evolution is only part of a statevector’s evolution. Under some circumstances (such as measurement) the “collapse” postulate is to be applied: when the states |ψi〉 become sufficiently