The origin of spontaneous localisation: Trace Dynamics II

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 2, 2019

The origin of spontaneous localisation: Trace Dynamics II

Tejinder Singh

As we saw earlier, trace dynamics is a matrix dynamics with a global unitary invariance, which operates at the Planck scale. Now we ask this question: suppose we do not wish to examine the dynamics on time scales of the order of Planck time or even smaller times, and we coarse-grain the trace dynamics over time steps much larger than Planck times, what is this time-averaged dynamics? It is shown, using the techniques of statistical mechanics, that this averaged dynamics is quantum (field) theory. No specific fine-tuning is done in the underlying trace dynamics so as to ensure that the emergent dynamics is quantum dynamics. Of central importance is the fact that as a  consequence of the global unitary invariance of the trace Lagrangian, trace dynamics possesses a conserved charge, the Adler-Millard charge. This charge has dimensions of action, and its equipartition as a result of doing the statistical mechanics is what is responsible for the emergence of quantum theory. It is plausible to assume that since the emergent theory results from averaging over time intervals much larger than Planck time, the energy scale associated with the emergent quantum system is much smaller than Planck energy. And this is of course true for the systems we currently study in laboratories while applying quantum field theory to the standard model of particle physics. It is already implicit in this analysis above that the dynamical laws on the Planck scale are not those of quantum field theory, but those of trace dynamics. This proposal will become much more plausible when we incorporate gravity into trace dynamics, in the theory of Spontaneous Quantum Gravity.

The principles of statistical mechanics are generally employed to derive the properties and thermodynamic laws for macroscopic systems, starting from the laws of atomic theory. Central to these principles is the fact that a macroscopic system is made of an enormously many constituent particles, whose individual motions we know how to describe, but we are not interested in. The macro-state is then determined by maximising the entropy made from all the microstates consistent with the values of physical attributes describing the macro-state (say constant temperature, or constant energy).

In the present instance of trace dynamics, these principles of statistical mechanics are being put to a different use, not for finding what happens to trace dynamics when we are considering a macroscopic system. Instead, we want to know what is the average dynamics of a system of one or more matrices obeying trace dynamics, when the dynamics is coarse-grained over time scales larger than Planck time scale. To find this average dynamics, we consider an ensemble of a very large number of matrices, each of which obeys trace dynamics at the Planck scale. Each one of them follows a different trajectory in the phase space, but we can assume [the ergodic hypothesis] that the ensemble average of the dynamics at any one time represents the long time average (i.e. the averaged dynamics of a matrix when coarse-grained over intervals larger than Planck time). We want to know the equilibrium ensemble, subject to the constancy of the trace Hamiltonian and the Adler-Millard charge.

One starts by defining a volume measure in the phase space made from the matrix elements - if there are N matrices in the trace dynamics, there is one canonical pair in the phase space for each of the elements of every one of the N matrices. A Liouville theorem is proved, namely that trace dynamics evolution preserves a volume measure in phase space. Next, the equilibrium density  distribution function in phase space is defined, which as usual gives the probability of finding the system in a given infinitesimal phase space volume. It is also shown that the ensemble average of the AM charge is a constant times a unit matrix, and since this constant has dimensions of action, it will eventually be identified with Planck’s constant subsequent to the emergence of the quantum dynamics.

The Boltzmann entropy is defined from the density distribution, it being determined by the number of matrix microstates consistent with a given constant value of the trace Hamiltonian and of the Adler-Millard charge. The equilibrium distribution is obtained by maximising the entropy subject to the constancy of these two conserved quantities. Next, we must recall that the energy scale we are interested in is much smaller than Planck scale; as a consequence the properties of the equilibrium distribution are determined by the AM charge, not by the trace Hamiltonian. A set of so-called Ward identities are proved for the equilibrium distribution, these being a generalised analog of the energy equipartition theorem in statistical mechanics. Important consequences follow from these identities. The AM charge is equipartitioned over all the bosonic and fermionic degrees of freedom and the equipartitioned value is identified with Planck’s constant. Recalling that the AM charge was defined in terms of the fundamental commutators and anti-commutators, it follows that the q, p degrees of freedom, when averaged over the equilibrium canonical ensemble, obey the commutation relations of quantum theory. Lastly, it is shown that the canonically averaged configuration variables and momenta obey the Heisenberg equations of motion of quantum theory. This last result follows because the (canonical averages of the) functions which define the time derivatives in the (first-order) Hamilton’s equations of trace dynamics get related to the canonically averaged commutators which eventually appear in the Heisenberg equations of motion. 

In this sense, quantum theory is an emergent phenomenon. When the equations of trace dynamics are coarse-grained over time intervals larger than Planck time, the emergent dynamics is quantum dynamics. This comes about because of the existence of the non-trivial Adler-Millard charge, unique to trace dynamics. The contact with quantum field theory is made by defining the Wightman functions of quantum field theory in terms of the emergent canonical averages of corresponding degrees of freedom in trace dynamics. One arrives at the standard relativistic quantum field theory for bosons and fermions. Since the Heisenberg equations of motion are now available, an equivalent Schrodinger picture dynamics can also be formulated.

Trace dynamics is one approach to derive quantum theory from symmetry principles, rather than arriving at quantum theory through the ad hoc recipe of `quantise the classical theory’. How can one justify the necessity of statistical mechanics in arriving at quantum theory? We recall that there is also an aspect of randomness/probabilities related to quantum mechanics - this being the aspect that comes into play during a quantum measurement. Randomness and probabilities are characteristic of statistical mechanics, specifically when fluctuations away from equilibrium become important. It is these fluctuations which are responsible for spontaneous localisation. We will take up this novel aspect of trace dynamics in the next post.