Quantum tunnelling in YUV

ABSTRACT

This paper derives and compares quantum-mechanical tunnelling rates in two frameworks: (i) the standard textbook picture (single-particle WKB, field-theoretic instantons, quantum fields on fixed curved backgrounds), and (ii) the YUV model in which a compact phase field φ is a potential-free, purely kinetic/topological field that carries energy–momentum and sources gravity, with topological vortices forming a Hubble-scale locked scaling network. We show that (a) to leading WKB/instanton order the textbook tunnelling exponentials remain the same unless the YUV phase field couples directly to the microsystem’s potential or unless local metric distortions induced by φ become comparable to microscopic barrier scales; (b) field-theoretic instanton/bounce methods generalize straightforwardly to include the φ energy-momentum if φ modifies the Euclidean background metric or boundary conditions; (c) vortices can produce localized metric perturbations (and thus small corrections to tunnelling exponents) but — under the minimal YUV coupling assumption (φ only sources gravity; no direct coupling to Standard Model fields) — these corrections are exponentially suppressed for microscopic barriers and only become relevant for macroscopic or cosmological tunnelling problems (early-universe phase transitions, vacuum decay across cosmological domain walls). We give explicit formulae for the tunnelling exponents (WKB and Euclidean action / bounce), compute first-order metric corrections for a spherically symmetric φ-induced perturbation, discuss instanton calculus in the presence of topological defects, and compare phenomenological consequences with standard results. Practical observational consequences and limits are discussed.

CONTENTS

Introduction

Standard WKB tunnelling (single particle) — review and formulas

Field theory tunnelling: Euclidean bounce and instantons (Coleman approach)

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Destek Ol

Quantum fields in curved space: basic framework and tunnelling generalization

YUV model ingredients and coupling assumptions

WKB with YUV: metric corrections to single-particle tunnelling

Field-theoretic instantons with YUV: bounce action in a φ-sourced metric

Vortex cores, topological tunnelling channels, and semiclassical rates

Comparison with Standard Model results and experimental implications

Conclusions and outlook

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References

INTRODUCTION

Quantum tunnelling is a cornerstone of quantum physics, with applications from α-decay to Josephson junctions and cosmological vacuum decay. In quantum mechanics the semiclassical WKB exponent controls tunnelling probabilities; in field theory the decay of metastable vacua is governed by Euclidean instanton (bounce) actions and pre-exponential determinants (Coleman et al.). [References: standard textbooks and reviews: Griffiths (intro WKB), Landau & Lifshitz (canonical QM), Coleman / instanton reviews].

The YUV model posits a compact phase field φ with purely kinetic Lagrangian (no potential), topological vortices, and the salient assumption used throughout this manuscript: φ is a real physical field carrying energy–momentum that enters Einstein’s equations as a source. The vortices form a Hubble-scale locked scaling network that provides a background topological energy density behaving like matter on cosmological scales (ρφ ∝ a⁻³ in matter era, in scaling attractor). In this note we quantify how that structure modifies standard tunnelling computations.

Plan: first summarize canonical WKB and instanton results (Sections 2–3), then include curvature (Section 4), introduce YUV ingredients and minimal coupling hypotheses (Section 5), then derive corrections (Sections 6–8), and finally discuss observational/experimental implications (Section 9).

STANDARD WKB TUNNELLING (SINGLE PARTICLE)

Consider a nonrelativistic particle of mass m moving in 1D potential V(x) with energy E < Vmax (barrier). The WKB transmission probability through a classically forbidden region [x1,x2] (turning points) is, in leading semiclassical order,

(2.1) 

T≈exp⁡(−2∫x1x2κ(x) dx),

where S_E[φ_b] is the Euclidean action of an O(4)-symmetric bounce (the lowest-action saddle point connecting false vacuum to true vacuum), and A is a one-loop determinant prefactor (det′). The bounce satisfies Euclidean equations of motion and appropriate boundary conditions (φ→ false vacuum at infinity) [Coleman]. This approach generalizes WKB to infinite degrees of freedom (instantons, bounces).

For a scalar field χ with potential U(χ) (false vacuum at χ=χ_f, true at χ=χ_t), the Euclidean action in flat space is

(3.2) 

SE[χ]=∫d4xE [12(∇Eχ)2+U(χ)].

Spherically symmetric ansatz χ = χ(ρ) with ρ = √(τ² + r²) reduces the problem to an ordinary differential equation for χ(ρ). The thin-wall analytic approximation is available when energy difference between vacua is small, giving an approximate action proportional to σ^4/(ΔU)^3 (σ = surface tension of bubble wall).

Instanton calculus and bounce determinants are standard — see Coleman and Paranjape (instanton techniques).

QUANTUM FIELDS IN CURVED SPACE: BASICS

Quantum field theory on curved backgrounds alters mode functions, vacuum definitions, and particle creation; the effective action includes curvature couplings (e.g., ξRχ²) and the background metric gμν appears in the action measure and derivatives. The relevant Euclidean action becomes

(4.1) 

SE[χ,g]=∫d4xEg [12gμν∇μχ∇νχ+U(χ)+12ξRχ2+… ].

Birrell & Davies provide the canonical exposition for field quantization in curved space and semiclassical gravity (⟨Tμν⟩ renormalization etc.). In the tunnelling context, two immediate effects occur: (i) the background metric changes the bounce solution and Euclidean volume measure; (ii) the metric itself can backreact because the bounce carries energy (so one should solve coupled χ–g equations if backreaction matters). For small gravitational effects, one can compute first-order corrections treating gravity perturbatively (Coleman–De Luccia formalism for vacuum decay including gravity).

YUV MODEL: ASSUMPTIONS AND MINIMAL COUPLING

We adopt the YUV assumptions you set for all subsequent discussions:

• φ is a compact phase field (φ ≡ φ + 2π), potential-free (V(φ)=0), with kinetic Lagrangian Lφ = −(f²/2) (∂μφ)(∂^μφ). (f is phase stiffness.)

• φ is a real physical field carrying energy–momentum; it contributes Tμν^{(φ)} to Einstein equations: Gμν = 8πG (Tμν^{baryon} + Tμν^{φ}).

• φ supports topological line defects (vortices) with quantized winding n, local gradient ∇φ ∼ n/r near core; energy density ρφ ∼ (f²/2)(∇φ)² ∼ f² n²/(2 r²). Vortex cores have finite regularized core radius r_c.

• On cosmic scales vortices form a Hubble-scale locked scaling network that supplies a background ρφ scaling like matter (on average, in matter era) and maintaining an effectively small, roughly constant fraction Ωφ of the total.

• Minimal coupling: We take φ to couple only gravitationally to Standard Model fields (no direct coupling g χ φ etc.). This is the minimal, conservative assumption; we will later comment on direct couplings and their stronger effects.

Under these assumptions, φ modifies tunnelling only via the metric (and possibly via boundary conditions if φ topological structures intersect the tunnelling region). Direct modifications of microscopic potential V(x) are absent by hypothesis.

WKB WITH YUV: METRIC CORRECTIONS TO SINGLE-PARTICLE TUNNELLING

Single-particle quantum tunnelling in curved space: the Schrödinger equation in a static curved metric for a nonrelativistic particle can be written using proper distance coordinate s with g_{ss} metric. For simplicity consider a static, spherically symmetric metric (isotropic coordinates, signature - + + +)

(6.1) 

ds2=−(1+2Φ(r))dt2+(1−2Ψ(r))dx⃗2,

with Newtonian potentials Φ,Ψ sourced by φ via Tμν^{φ}. For weak fields |Φ|,|Ψ| ≪ 1, the nonrelativistic Hamiltonian for a particle is corrected: the effective potential includes gravitational redshift:

(6.2) 

Veff(x)=Vmicro(x)+mΦ(x)+O(Φ2).

Hence the WKB exponent becomes

(6.3) 

ln⁡T≈−2∫s1s22mℏ2[Vmicro(s)+mΦ(s)−E] gss ds.

For small Φ, Taylor expand to first order: the leading correction to the exponent is

(6.4) 

δ(ln⁡T)≈−mℏ∫s1s2Φ(s)2m(V−E) ds,

neglecting metric factor corrections. Two primary conclusions:

A) If |Φ| on the barrier region is extremely small compared to V−E divided by m (typical microscopic barriers have V − E ~ eV–MeV while gravitational potentials Φ ~ 10⁻⁶ – 10⁻⁹ on galactic scales), the correction is utterly negligible. Thus for laboratory-scale tunnelling (Josephson junctions, α-decay), YUV gravitational corrections are effectively zero.

B) If tunnelling involves macroscopic barriers (cosmological domain walls, vacuum decay; or barriers with energy scales comparable to gravitational energy), then Φ can alter the action significantly and must be included.

Estimate: take m ~ proton mass, V − E ~ eV, barrier width L ~ nm, and |Φ| ≲ 10⁻⁸ (even near strong vortices), then δ(ln T) ≪ 1. Thus standard WKB stands for microscopic processes.

Important: if φ produces local strong gradients (for example inside a vortex core the local energy density is large), one must evaluate Φ by solving Poisson/Einstein eqns sourced by ρφ. We address this in Section 7.

(Reference for QM in weak curved backgrounds and matched WKB methods: Landau & Lifshitz; Griffiths; Birrell & Davies for field effects.)

FIELD-THEORETIC INSTANTONS WITH YUV: BOUNCE ACTION IN A φ-SOURCED METRIC

For field tunnelling (vacuum decay), include gravity by considering the Euclidean action for the tunnelling field χ plus φ contribution:

(7.1) 

SE[χ,g,ϕ]=∫d4xg {12gμν∂μχ∂νχ+U(χ)+12f2gμν∂μϕ∂νϕ}−116πG∫d4xg R.

We assume φ is classical (no potential) and that its background configuration contains vortices and a smooth average contribution ⟨(∂φ)²⟩. The bounce χ_b(ρ) solves coupled Euclidean χ and Einstein equations (with φ’s energy density entering Tμν). In practice we consider two regimes:

Regime I (weak backreaction): φ background energy density ρφ is small in the tunnelling volume (Ωφ small locally). Solve for χ bounce in fixed metric g = η plus small corrections computed perturbatively via linearized gravity. The Euclidean bounce action expands:

(7.2) 

SE=SE(0)[χ]+δSE[g(1)[ρϕ]]+⋯ ,

with δS_E linear in metric perturbation induced by φ. The leading correction to the exponent is small if 8πG ρφ L^2 ≪ 1 (L = characteristic bounce size). For microscopic bounces L ≪ (G ρφ)^{-1/2}, correction negligible.

Regime II (strong backreaction / large scales): If φ energy comparable to vacuum energy scales in the bounce region, one must solve fully coupled bounce + metric + φ field. This case is relevant for cosmological vacuum decay or phase transitions with bubble radii comparable to cosmic scales (e.g., nucleation across domain of vortices). Then φ can change both the shape of the bounce and the critical action drastically (Coleman–De Luccia style).

General formula (schematic) for the corrected tunnelling exponent to first order in δg:

(7.3) 

δSE≈∫d4xE 12δgμνTμν(χ)[χb]−116πG∫d4xE δ(gR)+⋯ .

Because Tμν^{(χ)} is localized near the bubble wall, the overlap with φ-induced δg localized in vortex regions controls the size of δS_E.

Reference frameworks: Coleman (bounce), Coleman & De Luccia (gravity), and Paranjape (instantons/extrapolations) for calculation methodologies.

VORTEX CORES, TOPOLOGICAL TUNNELLING CHANNELS, AND SEMICLASSICAL RATES

Unique to YUV: topological vortices are not just gravitational sources; they can act as geometric channels for tunnelling by locally altering both metric and the phase boundary conditions of fields coupled to φ (if any). Two distinct effects:

(i) Geometric modulation of barrier: A vortex region with elevated ρφ creates a local gravitational potential well (Φ < 0 for positive mass density), which either raises or lowers an external potential barrier from the particle’s perspective via mΦ term in Veff. For m>0 and Φ<0 the effective barrier can be slightly deeper, reducing tunnelling; sign and size depend on local φ configuration.

(ii) Topological boundary conditions: Because φ is compact and supports winding, fields with nontrivial φ-dependence might experience altered boundary conditions when the tunnelling path crosses a vortex. If some observable (e.g., a phase of a condensate or a gauge field holonomy) is tied to φ, then the effective action for a tunnelling path includes additional topological terms ~ ∮ φ dℓ or additional Berry phases. In our minimal coupling assumption (no direct coupling), this effect is absent. If direct microscopic coupling existed, it could either enhance or suppress tunnelling in a way that depends on winding n.

(iii) Collective tunnelling of topological defects: There is a separate class of tunnelling problems: nucleation or annihilation of vortices (topology change) itself. The action for such processes is computed by finding an instanton in the φ sector (a Euclidean configuration that interpolates between topologies). For the compact φ with Lagrangian Lφ = −(f²/2)(∂φ)², vortex nucleation costs an action of order f² times a geometric factor (roughly area of the nucleated loop times f² n²). Rates are then ∼ e^{−S_vortex/ħ}. These tunnelling problems are analogous to instanton calculations in Sine-Gordon or XY models (but note V(φ)=0, so the topological defects derive from compactness and boundary conditions rather than an explicit potential).

Instanton and semiclassical literature (instantons in gauge theories, topological defects) provides templates for computing such vortex nucleation rates; references include Coleman’s lectures and Paranjape’s instanton compilation.