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arXiv:2007.12025

On Dirac Quantisation rules and the trace anomaly

T. C. Choy

Published Jul 23, 2020• 0 citations

Abstract

In this article I shall clarify various aspects of the Dirac quantisation rules of 1930\cite{Dirac}, namely (i) the choice of antisymmetric Poisson brackets, (ii) the first quantisation Rule 1 (iii) the second quantisation Rule 2, and their relations to the trace anomaly. In fact in 1925 Dirac already had a preliminarily formulation of these rules \cite{Dirac3}. Using them, he had independently rediscovered the Born-Jordan quantisation rule \cite{BornJordan1925} and called it the quantum condition. This is the best known and undoubtedly most significant of the canonical quantisation rules of quantum mechanics. We shall discuss several violations of the Poisson-Lie algebra (assumed by Dirac), starting from antisymmetry, which is the first criterion for defining a Lie algebra. Similar violations also occur for the Leibniz's rule and the Jacobi identity, the latter we shall also prove for all our quantum Poisson brackets. That none of these violations jeopardised Dirac's ingenious original derivation \cite{Dirac} of his first quantisation Rule 1, is quite remarkable. This is because the violations are all of higher orders in $\hbar$. We shall further show that (ii) does not automatically lead to a trace anomaly for certain bounded integrable operators. Several issues that are both pedagogical and foundational arising from this study show that quantum mechanics is still not a finished product. I shall briefly mention some attempts and options to complete its development.

DOI: 10.13140/rg.2.2.16190.31041

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In this article I shall clarify various aspects of the Dirac quantisation rules of 1930\cite{Dirac}, namely (i) the choice of antisymmetric Poisson brackets, (ii) the first quantisation Rule 1 (iii) the second quantisation Rule 2, and their relations to the trace anomaly. In fact in 1925 Dirac already had a preliminarily formulation of these rules \cite{Dirac3}. Using them, he had independently rediscovered the Born-Jordan quantisation rule \cite{BornJordan1925} and called it the quantum condition. This is the best known and undoubtedly most significant of the canonical quantisation rules of quantum mechanics. We shall discuss several violations of the Poisson-Lie algebra (assumed by Dirac), starting from antisymmetry, which is the first criterion for defining a Lie algebra. Similar violations also occur for the Leibniz's rule and the Jacobi identity, the latter we shall also prove for all our quantum Poisson brackets. That none of these violations jeopardised Dirac's ingenious original derivation \cite{Dirac} of his first quantisation Rule 1, is quite remarkable. This is because the violations are all of higher orders in

\hbar

. We shall further show that (ii) does not automatically lead to a trace anomaly for certain bounded integrable operators. Several issues that are both pedagogical and foundational arising from this study show that quantum mechanics is still not a finished product. I shall briefly mention some attempts and options to complete its development.

In this paper we explore the idea of looking at the Dirac quantisation conditions as ¯h-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phase- space, we handle the above constraints using the standard theory of Dirac for constrained systems. The hope is to obtain, as total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets the Moyal ones. Unfortunately the program fails indicating that something is missing. We put forward at the end some ideas for future work which may overcome this failure. 1.

Heisenberg's breakthrough in his July 1925 paper that set in motion the development of Quantum Mechanics through subsequent papers by Born, Jordan, Heisenberg and also Dirac (from 1925 to 1927) is reexamined through a modern lens. In this paper, we shall discuss some new perspectives on (i) what could be the guiding intuitions for his discoveries and (ii) the origin of the Born-Jordan-Heisenberg canonical quantization rule. From this vantage point we may get an insight into Einstein's Quantum Riddle (Lande1974,Sommerfeld1918,Born1926) and a possible glimpse of what might come next after the last 100 years of Heisenberg's quantum mechanics.

We identify a contradiction within the condensate framework that forces second quantisation of the fermion sector. To our knowledge, no other interpretive programme derives the Born rule from dynamics independent of the field's algebraic structure, which is what makes this contradiction possible. The contradiction: Papers 1 and 3 independently established the Born rule (from competing Poisson processes) and tree-level QED cross-sections (from the coupled field equations). The optical theorem — whose application to the condensate's timelike excitations uses the standard relationship between the self-energy and forward attenuation — relates the total cross-section (positive, from the derived Born rule) to −2 Im Π(q²). The c-number Dirac field of Papers 2–3 produces identically zero vacuum polarisation and cannot satisfy this constraint. Among the standard quantisation options (c-number, bosonic, fermionic), only Grassmann promotion of the Dirac field resolves the contradiction: it produces vacuum fluctuations, a nonzero Im Π with the correct sign, and positive detection rates. Bosonic promotion gives negative spectral weight; parastatistics of order p > 1 violates the magnitude constraint. The correct statistics for spins 0, ½, and 1 is thereby selected, at the level of a one-loop necessary condition: half-integer spin requires anticommutation, integer spin requires commutation, and in each case the wrong statistics produces detection rates inconsistent with the competition mechanism. The implementation uses standard second-quantised machinery (Fock vacuum, anticommutation relations, Feynman propagator); what is new is the framework-internal argument that establishes the necessity of this machinery for fermions.

Abstract This chapter covers the process of the consolidation of the ideas in Heisenberg’s Umdeutung paper into the Two-Man-Paper by Born and Jordan, the Three-Man-Paper (Dreimännerarbeit) by Born, Heisenberg, and Jordan, and a paper by Dirac. Both Born and Jordan recognized that Heisenberg’s peculiar non-commutative multiplication rule is recognized to be nothing but the standard multiplication rule for matrices. They also replaced Heisenberg’s quantization condition, the Thomas–Reiche–Kuhn sum rule, by the now-familiar commutation relations of position and momentum. The chapter then discusses how Dirac independently derived these commutation relations, relying on a profound analogy with the Poisson brackets of classical mechanics