The origin of spontaneous localisation: Trace Dynamics I

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; Matrix Models

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December 1, 2019

The origin of spontaneous localisation: Trace Dynamics I

Tejinder Singh

We have seen earlier that spontaneous localisation is a falsifiable but ad hoc modification of the Schrodinger equation, for explaining the absence of macroscopic position superpositions. One would like to derive this collapse theory from an underlying dynamics, based on a symmetry principle. One such theory, known as Trace Dynamics (TD) has been developed by Stephen Adler (IAS, Princeton) and collaborators. The best source to read about this theory is Adler’s book `Quantum theory as an emergent phenomenon’ [Cambridge University Press, 2004].

Put in brief, Trace Dynamics is the dynamics of matrix models which obey a global unitary invariance. Quantum (field) theory is emergent as the statistical mechanics of these matrix models. TD is assumed to operate at the Planck scale, although space-time is assumed to be Minkowski space-time. The matrix models describe the dynamics of fermionic matter, as well as of gauge fields, although no specific form of the Lagrangian of the theory is prescribed. Gravity is not included.

What is the motivation behind trace dynamics? One would not like to arrive at quantum theory by quantising a classical theory. This is considered unsatisfactory because the classical theory is only a limiting case of quantum theory - one should not have to know the limit of a theory to construct the theory; it should be the other way around. [We do not construct special / general relativity by `relativising’ Newtonian mechanics / gravitation. The relativity theories are built from their own symmetry principles, and yield the Newtonian theories in the limit]. Trace dynamics is a first principles theory, from which quantum, and classical mechanics, are emergent as approximations.

In TD, particles and fields are not described by real numbers, but by matrices (equivalently operators). Consider for instance the Newtonian Lagrangian dynamics of a collection of particles with configuration variables q_i . From these variables and their time derivatives we can make the Lagrangian, and from there obtain the equations of motion. To construct trace dynamics, assume instead that each of the q_i is a matrix/operator. We will construct its corresponding velocity by taking the time derivative of the matrix, which is equivalent to taking time derivative of each matrix element. Given this set of configuration matrices and their velocities, the Lagrangian of TD is constructed by making a polynomial from matrix products, and then taking the matrix trace of this polynomial. Thus the Lagrangian is a scalar (a real number), rightfully called the trace Lagrangian. The action in TD is the time integral of this trace Lagrangian. Lagrange equations of motion are obtained by extremising the action by varying it with respect to the operator configuration variables. This requires the introduction of a `trace derivative’ - which is a natural method for differentiating  trace of a polynomial w.r.t. an operator. The resulting Lagrange equations are operator equations. Also, TD is constructed to be a Lorentz invariant theory. Importantly, the configuration variables and their conjugate momenta, all non-commute with each other. Unlike in quantum theory, the commutators in TD are arbitrary and time-dependent, in general.

What, one might ask, is the point of these matrices/operators? What is the physical interpretation of their matrix elements? TD is a pre-quantum theory, more general than quantum theory, operating at the Planck scale; a theory from which quantum mechanics will be derived as an emergent approximation, at energy scales below the Planck scale. The eigenvalues and eigenstates of these matrices represent possible values that these degrees of freedom can take, and superposition  is possible too; yet the rules of evolution are not those of quantum theory.

Trace dynamics can be extended to fields too, by dividing three-space into cells and assigning one q-operator for every cell, which then represents the field value in that cell. Alternatively, one can take the continuum limit of the N-particle trace dynamics. 

The elements of the matrices in TD are made from complex numbers and complex Grassmann numbers. A complex Grassmann number is made from two real Grassmann numbers. A real Grassmann number is made from products of Grassmann elements. Grassmann elements anti-commute with each other, unlike ordinary numbers which commute with each other. Thus the square of a Grassmann number is zero. A product of an even number of Grassmann elements commutes with every Grassmann element, and together with unity these form an even-grade Grassmann algebra, which is used to represent bosonic fields. A product of an odd number of Grassmann elements anti -commutes with other odd number products of Grassmann elements: together these form the odd-grade sector of the Grassmann algebra. These are used to represent fermionic fields. A general matrix made from complex Grassmann numbers can be written as a sum of two matrices, one made from even-grade elements (and called bosonic) and one made from odd-grade (and called fermionic).

In matrix dynamics the trace Hamiltonian of the system is conserved, as can be expected. In addition though, trace dynamics possesses a fascinating conserved charge, which results from the global unitary invariance of the trace Lagrangian and trace Hamiltonian. This charge is known as the Adler-Millard charge, it has no analog in ordinary classical dynamics, and is responsible for the emergence of quantum theory from trace dynamics. It is given by the sum (over bosonic degrees of freedom) of the commutators [q,p], minus the sum (over fermionic degrees of freedom) of the anti-commutators {q,p}. Note that this charge has the dimensions of action. If the Hamiltonian  is self-adjoint, the AM charge is anti-self-adjoint. If the Hamiltonian were to also have an anti-self-adjoiint piece, the AM charge picks up a self-adjoint component. One can also define a generalised Poisson bracket in trace dynamics, and express the dynamics as Hamilton’s equations of motion, and also in terms of Poisson brackets. In general, evolution in trace dynamics is not unitary, and differs from the evolution given by the Heisenberg equations of motion in quantum theory.

Trace dynamics in many ways resembles matrix models that have been studied in quantum field theory. However, in matrix models, canonical quantum commutation relations are imposed on the matrix elements, thus leading to the standard quantisation. Unlike matrix models though, one does not quantise a trace dynamics. Instead one asks, what does the coarse grained dynamics look like, if trace dynamics is averaged over time-intervals much longer than Planck time, and one is interested in the emergent dynamics at energy scales much smaller than Planck scale? It turns out that this emergent dynamics is quantum dynamics. This is established by applying the conventional techniques of statistical mechanics to trace dynamics, a topic which we will take up in the next post.