Massive gauge boson and Higgs boson as the vestiges of non-Fock vacuum

A microscopic model of the Brout-Englert-Higgs (BEH) mechanism is proposed. Massless fermions and antifermions do not belong to the Fock space with definite particle-number distribution, but belong to a non-Fock space with indefinite one. From this vast space, their ground-state is selected by a kinematical condition. Due to the interaction between them, the Fock state in which fermions and antifermions are massive is restored, but this state has the vestiges of the non-Fock state in the massive gauge boson and the Higgs boson. In the non-Fock state, massless fermions and antifermions exist as pairs, and behave as quasi bosons within a small space-time region. Due to Bose statistics, the direction of their center-of-mass motion is parallel to each other, and their transverse excitations are suppressed by an energy gap, making the gauge boson coupled to them massive. The Higgs boson is not an elementary particle, but a quasiparticle appearing after the Fock vacuum is restored.

💡 Research Summary

The paper proposes a novel microscopic picture of the Brout‑Englert‑Higgs (BEH) mechanism in which the origin of gauge‑boson masses and the Higgs particle is traced back to a “non‑Fock vacuum”. The author starts from a chiral‑symmetric Lagrangian containing a massless Dirac fermion ϕ coupled to a U(1) gauge field Bμ. Because the fermion is massless, the usual Fock construction with a definite particle‑number distribution is argued to be inappropriate. Instead, the vacuum is taken to be a non‑Fock state with an uncountable basis, i.e. a space of indefinite particle number.

A key step is the introduction of a kinematical condition that treats the annihilation of a fermion and the creation of an antifermion as equivalent operations when the particles are massless. This leads to new mixed raising and lowering operators (e_a(p)=\cos\theta_p,a(p)+\sin\theta_p,b^\dagger(-p)) and (e_b(-p)=\cos\theta_p,b(-p)-\sin\theta_p,a^\dagger(p)). The angle θp depends on momentum: at p→0 one has cosθ=sinθ, while at large p the antifermion component disappears (sinθ→0). Using these operators the author builds a vacuum |e₀⟩ that is a direct product over all momenta of superpositions of the empty Fock vacuum and fermion‑antifermion pair excitations weighted by sinθp. This state lives in a non‑Fock Hilbert space of continuous cardinality.

To connect this exotic vacuum to the observed world, a mean‑field approximation is employed. A scalar mean field U₀ is introduced, modifying the fermion kinetic term to (\bar\phi(i!\not!\partial+U_0)\phi). The diagonalisation condition fixes θp through the relation (\cos2\theta_p = \frac{1}{2}\bigl(1+\frac{\epsilon_p}{\sqrt{\epsilon_p^2+U_0^2}}\bigr)), with εp the fermion energy. This mean field restores a conventional Fock vacuum for massive fermions, but the “vestiges” of the original non‑Fock structure remain encoded in the gauge sector.

The paper argues that in the non‑Fock vacuum massless fermion‑antifermion pairs behave as quasi‑bosons within a limited space‑time region. Their composite operators (P_k = b(-k,\downarrow)a(k,\uparrow)) obey Bose‑like commutation relations for different momenta, while Pauli exclusion modifies the algebra for identical momenta. Because of Bose statistics, transverse excitations of these pairs acquire an energy gap, rendering the gauge boson that couples to them massive. This mechanism is presented as the microscopic origin of the vacuum expectation value v_h that appears in the usual Higgs potential.

Beyond the mean‑field level, collective excitations of the fermion‑antifermion pairs generate a scalar quasiparticle identified with the Higgs boson. Its mass m_H emerges from the same parameters (U₀, λ, v_h) that define the Higgs potential, suggesting that the Higgs is not an elementary scalar but a bound‑state‑like mode arising after the non‑Fock vacuum is “restored”.

The author discusses several implications: (a) the non‑Fock vacuum provides a new perspective on spontaneous symmetry breaking, (b) the Higgs particle is reinterpreted as a quasiparticle rather than a fundamental field, and (c) experimental signatures might include residual effects of the quasi‑boson pairs in high‑energy processes. However, the paper leaves many technical issues open. The construction of the non‑Fock vacuum, its normalization, and the preservation of unitarity are not rigorously demonstrated. The mean‑field approximation is invoked without a systematic justification, and the connection to the full non‑abelian electroweak sector is only sketched. Moreover, the proposed mechanism must reproduce the precise electroweak precision observables that the standard BEH framework already fits.

In summary, the work offers an imaginative attempt to derive gauge‑boson masses and the Higgs boson from a vacuum structure that lies outside the conventional Fock space. While the conceptual novelty is noteworthy, substantial further work is required to establish mathematical consistency, to embed the idea into the full Standard Model, and to identify concrete experimental tests that could distinguish this scenario from the standard Higgs mechanism.