Searcharxiv⌕ Search

arXiv subjects

Dan Solomon

Publications and source records attributed to Dan Solomon.

At least 19 recordsLinked to original sources
arXiv2023

Spectral Theorems for Generalized Weyl Nodes with Impurities in a Magnetic Field

We prove a few spectral theorems for the density of states of a Weyl node with arbitrary topology. We show that the density of extended states of a Weyl node with random impurity potentials remains gapless in the presence of a magnetic field. Therefore, a magnetic field precludes Anderson localization in Weyl semi-metals, when inter-node transitions are suppressed for smooth enough potentials. We also provide a rigorous quantum mechanical proof of the chiral magnetic effect for arbitrary topology of a Weyl node.

cond-mat.str-el↗
arXiv2017

The Vacuum Polarization tensor in 1+1 dimensional space-time

Quantum field theory (QFT) is supposed to be gauge invariant. However it has been well established that a direct calculation of the vacuum polarization tensor produces a non-gauge invariant result. In this paper it will be shown that this problem is due to the fact that there is a lower bound to the free field energy in QFT. The vacuum polarization tensor will be calculated in 1+1 dimensional space-time and is shown not to be gauge invariant. The gauge invariance of the theory can be restored through a regularization procedure which eliminates the non-gauge invariant terms. However it will be shown that this will impact the free field energy. If the free field energy is defined so that the vacuum state has an energy of zero then the impact of regularization is to introduce states whose free field energy is less than zero.

physics.gen-ph↗
arXiv2016

Taming the divergent terms that occur during adiabatic switching in perturbation theory

A potential problem with adiabatic switching in perturbation theory is that divergent terms appear in the series solution. An example of this was presented by C. Brouder et al [4] for a simple 2 state system where the evolution of system in the presence of a time dependent perturbation was considered. One of their results is that the evolution operator has no well-defined limit for adiabatic switching. We will rework this problem to show that for adiabatic switching the evolved states are well-defined with any divergences being absorbed in a time independent phase factor which can be removed. These results will then be applied to the more general problem of a system with an arbitrary number of states. It will be shown that for this case, also, the potentially divergent terms all appear in a time-independent phase factor.

quant-ph↗
arXiv2015

Why do non-gauge invariant terms appear in the vacuum polarization tensor?

It is will known that quantum field theory at the formal level is gauge invariant. However a calculation of the vacuum polarization tensor will include non-gauge invariant terms. These terms must be removed from the calculation in order to get a physically correct result. One common way to do this today is the technique of "dimensional regularization". It has recently been noted [2] that at one time a supersymmetric-like solution to the problem was explored - that is, the right combination of fields would cause the offending terms to cancel out. I will examine some of this early work and pose the question - why do the non-gauge invariant terms appear in the first place? I will show that this is due to an improper mathematical step in the formulation of the pertubative expansion. I will then show that when this step is corrected the result is gauge invariant. However a supersymmetric-like solution is still required to cancel out a divergent term.

physics.gen-ph↗
arXiv2014

The expectation value of the field operator

Much of the mathematical development of quantum field theory has been in support of determining the S-matrix in order to calculate scattering cross sections. However there is also an interest in determining how expectation values of field operators evolve in time from an initial state. In this paper I will examine some aspects of this problem.

physics.gen-ph↗
arXiv2014

Deriving Z[J] from the time evolution operator

An important quantity in quantum field theory is the vacuum-to-vacuum transition amplitude in the presence of an external source. This quantity is often designated by Z[J] and is the generator for the n-point Greens functions. In textbooks Z[J] is often derived using a path integral formulation, however it will be shown here that it is possible to derive Z[J] directly from the time evolution operator without using the path integral formulation.

quant-ph↗
arXiv2013

Second quantization and gauge invariance

It is well known that the single particle Dirac equation is gauge invariant. This means that observable quantities, such as the current density, are not affected by a gauge transformation. However what happens when the method of second quantization is applied to convert a single particle theory into a field theory? In this case it will be shown that the theory is no longer gauge invariant. This will be shown by considering the second quantization of a zero mass Dirac field in 1+1 dimensions and examining the change of the current density operator due to a gauge transformation.

quant-ph↗
arXiv2013

A Comment on the calculation of the Lie algebra cocycle in the book Loop Groups

An expression for the Lie algebra cocycle corresponding to the central extension of the group GLres(H) is derived in the book Loop Groups [1]. It will be shown in this paper that some of the terms in this expression are ambiguous and in order to be evaluated correctly the trace operation must be properly defined.

math-ph↗
arXiv2013

A demonstration of the necessity of regularization in order to avoid inconsistent results in quantum field theory

We will examine a particular mathematical derivation in a paper by P. Falkensteiner and H. Grosse (F&G) [1]. In [1] a quantity "delta(A)" is defined. This quantity is generated when the normal ordered generalized charge operator undergoes a unitary transformation. Using standard mathematical techniques F&G convert "delta(A)" from its original form to another form which is suppose to be equivalent. It will be shown here that the two forms are not equivalent and that there is a mathematical inconsistency in their derivation. We will examine the source of this inconsistency and show that it can be resolved by proper regularization of the mathematical expressions.

quant-ph↗
arXiv2012

Relating the solutions of the Dirac equation with a background electric potential to solutions with a background pseudoscalar potential

We compare two different solutions of the Dirac equation in (1+1) dimensions. One solution is for a fermion in the presence of an electric potential and the other is for a fermion in the presence of a pseudoscalar potential. It is shown that for properly defined potentials one can easily relate the solutions of one system to the solutions of the other. In effect, solving one problem gives the solution to both. In addition, the vacuum charge density is calculated in the both cases and it is shown how this result is impacted by the presence of anomalies in quantum field theory.

quant-ph↗
arXiv2012

The effect of point split regularization on the sign of the Casimir energy

In a recent paper [1] the Casimir energy was calculated for a massive dirac field in (1+1) dimensional space-time in the presence of an inverse square well potential and shown to be positive. It will be shown that this result violates a key assumption of quantum field theory which is that the vacuum state is the state of minimum energy. The reason for this discrepancy is examined and is shown to be related to the way the charge density operator is defined. If the charge density operator is defined using point splitting then an extra term will be added to the charge which will result in the Casimir energy being negative.

quant-ph↗
arXiv2012

The vacuum state and minimum energy in Dirac's hole theory

In Dirac's hole theory the vacuum state is assumed to be the state where all negative energy states are occupied and all positive energy states are unoccupied. This is often referred to as the Dirac sea. It is generally assumed that the Dirac sea is the minimum possible energy state. However it will be shown in this paper that this is not the case.

physics.gen-ph↗
arXiv2011

Comparing two methods of regularization of the kinetic energy density

In this paper we will compare two different methods of regularizing the kinetic energy density for a massless scalar field in the presence of a static scalar potential. One method of regularization is to subtract the cosmological constant from the "naive" expression for the kinetic energy density. The other method is to use point split regularization. It is found that the two methods yield different results. The result obtained using point split regularization includes an extra ambiguous term.

gr-qc↗
arXiv2011

An example of a violation of the spatial quantum inequality with a comment on the quantum interest conjecture

It is generally known that the energy density can be negative in quantum field theory. It is also believed that there are limits on this negative energy density. These limits are known as the quantum inequalities. In a recent paper [8] an example was provided of a system which violated the quantum inequalities. Here we will demonstrate a violation of the spatial quantum inequality for a scalar field with zero mass in 1-1 dimensional space-time. In addition we will show that the system presented here also violates the quantum interest conjecture.

quant-ph↗
arXiv2011

Some ambiguities with point split regularization and its impact on a proof of the spatial quantum inequality

In classical physics the energy density of a field is always positive. However this does not hold true for quantum physics where the energy density of a field can be locally negative. There are limits on the weighted average of this negative energy density called the quantum inequalities. Recently this author has provided a number of examples which show that the quantum inequalities are not valid. In this paper we will examine a previously published proof of the spatial quantum inequality for a zero mass scalar field in 1-1 dimensional space-time. It will be shown that there is a possible problem with this proof due to an ambiguity associated with point split regularization and the definition of the Hadamard form for the two point function.

quant-ph↗
arXiv2010

A violation of the spatial quantum inequality

In classical physics the energy density of a field, such as the electromagnetic field, is always positive. However, in quantum field theory it has been shown that the energy density can be negative. There are restrictions, called the quantum inequalities, on the amount of negative energy that can exist in some region of space and time. In this paper we will focus on the spatial quantum inequality as it applies to a massless scalar field in 1-1 dimensional space-time. The spatial quantum inequality is a restriction on the amount of negative energy that can exist in a region of space at a given time. It will be shown that we can specify a quantum state which violates the spatial quantum inequality.

quant-ph↗
arXiv2010

A counter-example to the quantum interest conjecture

According to the quantum interest conjecture any negative energy pulse must be associated with a positive energy pulse of greater magnitude than that of the negative energy pulse. In this paper we will demonstrate a counter-example to this conjecture. We will show that, for a massless scalar field in 1-1D space-time, it is possible to generate an "isolated" negative energy pulse.

quant-ph↗