The origin of spontaneous localisation: Trace Dynamics III

SCEST21: Schrodinger's Cat, and Einstein's Space-time, in the 21st Century

A blogspot for discussing the connection between quantum foundations and quantum gravity

Managed by: Tejinder Pal Singh, Physicist, Tata Institute of Fundamental Research, Mumbai

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Keywords: Quantum foundations; Quantum gravity; Schrodinger's cat; Spontaneous collapse theory; Trace Dynamics; Statistical Mechanics
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December 3, 2019

The origin of spontaneous localisation: Trace Dynamics III

Tejinder Singh

We have seen that the theory of trace dynamics gives rise to relativistic quantum (field) theory as an emergent phenomenon, after one constructs the equilibrium statistical thermodynamics of the underlying theory. This emergence opens up the possibility that spontaneous localisation is also a consequence of trace dynamics, in the following sense. When one arrives at the thermodynamic approximation by constructing the equilibrium configuration by applying statistical mechanics to the underlying microscopic theory, it is assumed that statistical  fluctuations away from equilibrium are negligible. Under certain circumstances though, fluctuations could become important. A situation of precisely this kind gives rise to spontaneous localisation, starting from the underlying trace dynamics, subject to certain assumptions. These assumptions will be justified when we incorporate gravity into trace dynamics (to be discussed in subsequent posts). 

Recall that the Adler-Millard charge is anti-self-adjoint, whereas the trace Hamiltonian is assumed to be self-adjoint.  We wish to consider the impact of statistical fluctuations on the emergent quantum dynamics. The description of these  fluctuations, and the stochastic correction that they imply to the trace Hamiltonian, is controlled by the Adler-Millard charge.  While the averaged Adler-Millard charge is equipartitioned, and proportional to Planck’s constant times a unit imaginary matrix, the fluctuations need not be equipartitioned. Moreover, one can by hand assume that the fluctuations can have an imaginary part, thus resulting in the Hamiltonian having an anti-self-adjoint part. [In the theory of spontaneous quantum gravity, this does not have to be put in by hand - the fundamental Hamiltonian at the Planck scale naturally has an anti-self-adjoint part]. 

The inclusion of imaginary fluctuations around equilibrium paves the way for spontaneous localisation to arise, provided some additional assumptions are made. It is assumed that such localisation takes place only for fermions (i.e. only in the matter sector, not for bosonic fields). Furthermore, only the non-relativistic case is considered, I.e. the Schrodinger equation for matter particles. Stochastic linear terms are added to the Hamiltonian, to represent fluctuations about equilibrium. Since the fluctuations include an imaginary part, the evolution of the modified Schrodinger equation does not preserve the norm of the state vector. [Norm preservation is essential, if one is to derive the Born probability rule]. The requirement of norm preservation is sought to be justified on empirical grounds, because particle number is conserved in the non-relativistic theory, and is related to the norm. Hence, a new state vector - which preserves norm - is defined by scaling the old state vector by its  norm. This makes the modified Schrodinger equation into a stochastic non-linear differential equation, which with the further assumption of no superluminal signalling, acquires precisely the form as in the GRWP theory of spontaneous collapse. And collapse does take place at a faster rate when more and more particles are entangled with each other. Entanglement enhances spontaneous collapse, making the equilibrium unstable. In fact, we have to recall that the mean dynamics arose after averaging over time scales larger than Planck time. Precisely this assumption - that such an averaging can be done - breaks down for macroscopic systems, as we will see in spontaneous quantum gravity! Moreover, there we will also find justification for the assumptions made above.

The take away note from this post is that trace dynamics contains within itself the roots for explaining spontaneous collapse, because quantum dynamics in the first place arises as a statistical thermodynamics approximation to the underlying matrix dynamics. There is every cause for investigating circumstances where fluctuations become important. And the classical world arises precisely because of such circumstances. Trace dynamics is presently the only known theory which provides a theoretical basis for spontaneous collapse.

A major limitation of trace dynamics is that while it operates at the Planck scale, it assumes space-time to  be Minkowski space-time. This however is clearly only a `transitory’ assumption, meant to be eventually relaxed. Until recently, it was not clear how to incorporate gravity as a matrix dynamics. We now know that Alain Connes’ non-commutative geometry programme enables us to do that. In so doing, we will also see how various assumptions made in trace dynamics get justified. 

Incorporating gravity into trace dynamics finally helps us understand that spontaneous localisation arises because the fundamental Hamiltonian at  the Planck scale is not self-adjoint. It does not have to be. Only the emergent Hamiltonian in quantum theory has to be self-adjoint. We will also see how we arrive at a formulation of quantum theory which does not depend on classical space-time: this was one of our stated goals. We will also have a theory of quantum gravity which dynamically explains absence of superposition of classical space-time geometries. The theory also explains the origin of black hole entropy, and suggests a quantum gravitational origin for dark energy.

In the next post, we provide an overview of spontaneous quantum gravity, and in subsequent posts we discuss the theory in some detail.