[Paper Review] The Anomalous Nambu-Goldstone Theorem in Relativistic/Nonrelativistic Quantum Field Theory

Tadafumi Ohsaku|arXiv (Cornell University)|Dec 1, 2013

Noncommutative and Quantum Gravity Theories68 references3 citations

TL;DR

This paper establishes a generalized counting law for Nambu-Goldstone (NG) bosons in nonrelativistic and Lorentz-symmetry-violating quantum field theories, showing that the number of true NG bosons is reduced by the rank of the commutator algebra of conserved charges. It identifies a quasi-Heisenberg algebra in the NG sector, implying modified quantum uncertainty relations and suggesting physics beyond standard quantum mechanics, with connections to number theory and the Riemann hypothesis.

ABSTRACT

The anomalous Nambu-Goldstone (NG) theorem which is found as a violation of counting law of the number of NG bosons of the normal NG theorem in nonrelativistic and Lorentz-symmetry-violated relativistic theories is studied in detail, with emphasis on its mathematical aspect from Lie algebras, geometry to number theory. The basis of counting law of NG bosons in the anomalous NG theorem is examined by Lie algebras (local) and Lie groups (global). A quasi-Heisenberg algebra is found generically in various symmetry breaking schema of the anomalous NG theorem, and it indicates that it causes a violation/modification of the Heisenberg uncertainty relation in an NG sector which can be experimentally confirmed. The formalism of effective potential is presented for understanding the mechanism of anomalous NG theorem with the aid of our result of Lie algebras. After an investigation on a bosonic kaon condensation model with a finite chemical potential as an explicit Lorentz-symmetry-breaking parameter, a model Lagrangian approach on the anomalous NG theorem is given for our general discussion. Not only the condition of the counting law of true NG bosons, but also the mechanism to generate a mass of massive NG boson is also found by our examination on the kaon condensation model. Furthermore, the generation of a massive mode in the NG sector is understood by the quantum uncertainty relation of the Heisenberg algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in the effective potential of the kaon condensation model. Hence the relation between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode coupling, and the mechanism of mass generation in an NG sector is established. Finally, some relations between the Riemann hypothesis and the anomalous NG theorem are presented.

Motivation & Objective

• To resolve the long-standing controversy over the counting law of NG bosons in nonrelativistic and Lorentz-symmetry-violated systems.
• To clarify the mechanism behind the anomalous Nambu-Goldstone theorem, where the number of NG bosons does not match the number of broken generators.
• To establish a geometric and algebraic framework using Lie algebras, Lie groups, and effective field theory for understanding the anomalous behavior.
• To explore connections between the anomalous NG theorem and deep mathematical structures, including class field theory and the Riemann hypothesis.
• To propose a modified formulation of the NG theorem that accounts for massive modes and quantum uncertainty effects in the NG sector.

Proposed method

• Derives a generalized counting law using the commutator algebra of conserved charges: $ n_{\text{true-NG}} = n_{BS} - \frac{1}{2} \text{rank}\langle [Q^A, Q^B] \rangle $.
• Analyzes the structure of the NG sector via Lie algebra and Lie group representations, identifying a generic quasi-Heisenberg algebra in symmetry breaking schemes.
• Applies the effective potential formalism to model the mechanism of anomalous NG behavior, particularly in a bosonic kaon condensation model with finite chemical potential.
• Uses differential geometry and Cartan connections to describe the vacuum manifold and its submanifold structure, linking topology to mode counting.
• Introduces a generic Lorentz-violating Lagrangian to model the anomalous NG theorem across different symmetry breaking patterns.
• Establishes a correspondence between the vacuum degeneracy structure and class field theory, including cyclotomic extensions and ideles/adeles, via the Kronecker-Weber theorem.

Experimental results

Research questions

• RQ1 Why does the number of NG bosons deviate from the number of broken generators in nonrelativistic and Lorentz-symmetry-violating systems?
• RQ2 How is the mass generation of NG modes explained in the anomalous NG theorem, particularly in the absence of explicit symmetry breaking?
• RQ3 What is the role of the Heisenberg algebra and quantum uncertainty in modifying the standard NG theorem in the anomalous regime?
• RQ4 How can the anomalous NG theorem be unified across relativistic and nonrelativistic frameworks using Lie group and algebraic geometry?
• RQ5 What are the deep mathematical connections between the anomalous NG theorem and number theory, especially the Riemann hypothesis?

Key findings

• The number of true NG bosons is given by $ n_{\text{true-NG}} = n_{BS} - \frac{1}{2} \text{rank}\langle [Q^A, Q^B] \rangle $, where the rank of the charge commutator matrix determines the number of massive modes.
• A quasi-Heisenberg algebra emerges generically in the NG sector, implying a modification of the Heisenberg uncertainty relation that could be experimentally detectable.
• In the kaon condensation model with finite chemical potential, the effective potential reveals a mechanism for generating massive modes via quantum fluctuations and mode-mode coupling.
• The vacuum manifold structure leads to a cyclotomic extension, which is an Abelian extension, and connects to class field theory via the Kronecker-Weber theorem.
• The paper establishes a formal link between the anomalous NG theorem and the Riemann hypothesis through the use of $ l $-adic Galois representations and idele class fields.
• The traditional one-to-one correspondence between broken generators and massless NG bosons is violated; instead, the broken generator space is projected into a lower-dimensional subspace due to nontrivial commutator structure.