Volume 377, Issue 13, 17 May 2013, Pages 945–951

Mass generation in the aether: Neutrino oscillations and massive gauge fields

Roman Tomaschitz

Department of Physics, Hiroshima University, 1-3-1 Kagami-yama, Higashi-Hiroshima 739-8526, Japan

Received 29 January 2013

Accepted 13 February 2013

Available online 14 February 2013

Communicated by V.M. Agranovich

Abstract

Neutrino mixing is studied in an absolute spacetime conception based on a dispersive aether. The effect of the frequency-dependent permeability of the aether on the interference phase of neutrino mass eigenstates is analyzed. Neutrinos are treated as massless Dirac spinors, and mass eigenstates are due to the neutrino permeability of spacetime. The aether can also generate effective gauge masses, resulting in massive dispersion relations preserving the gauge symmetry. The propagators of gauge and spinor fields are derived, illustrating mass generation by isotropic permeability tensors in the aether frame, the rest frame of the cosmic background radiation.

Highlights

► Dirac neutrinos and gauge fields couple by permeability tensors to the aether. ► Effective neutrino and gauge masses are generated by permeability tensors. ► Oscillating flavor transition probabilities are due to effective mass-squares. ► The propagators of Dirac and gauge fields in a dispersive spacetime are derived. ► Gauge invariance is preserved by the effective mass in the propagators.

Keywords

Dispersive spacetime;
Mass eigenstates of Dirac fermions;
Neutrino mixing in a permeable spacetime;
Effective neutrino mass and flavor oscillations;
Effective mass-squares of gauge fields;
Michelson–Morley experiments with neutrino beams

1. Introduction

We investigate the generation of mass in the dispersion relations of Dirac particles and gauge fields. Neutrino masses required for flavor oscillations as well as massive non-Abelian gauge fields are obtained by coupling the massless wave equations to permeability tensors. Flavor mixing [1], [2], [3], [4] and [5] is compatible with sub- as well as superluminal neutrino velocities [6], [7], [8] and [9], and arises in a dispersive spacetime due to the frequency-dependent permeabilities of the aether [10] and [11].

The aether defines a distinguished frame of reference, physically manifested as the rest frame of the isotropic cosmic microwave background radiation [12], [13], [14], [15] and [16], and wave fields couple to the aether with isotropic permeability tensors in this frame [17], [18], [19] and [20]. We discuss the permeability tensors of Dirac fermions and gauge bosons, the interference phase of neutrino mass eigenstates [1], [5] and [21], as well as gauge fixing and propagators in the dispersive aether, and potential Michelson–Morley experiments with neutrino beams.

Dirac particles freely propagating in the aether are described by the Dirac equation

equation1.1

γ_μg^μνψ_,ν+mψ=0

γ_{μ} g^{μ ν} ψ_{, ν} + m ψ = 0

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coupled to a real symmetric permeability tensor g^μν(ω)

g^{μ ν} (ω)

, which depends on the frequency of the spinor modes ψ∝exp(ik_μ(ω)x^μ)

ψ \propto \exp (i k_{μ} (ω) x^{μ})

[10]. The Dirac matrices satisfy γ_μγ_ν+γ_νγ_μ=2η_μν

γ_{μ} γ_{ν} + γ_{ν} γ_{μ} = 2 η_{μ ν}

, where γ₀

γ_{0}

is anti-Hermitian, the γ_i

γ_{i}

are Hermitian, and the sign convention for the Minkowski metric is η_μν=diag(−1,1,1,1)

η_{μ ν} = diag (- 1, 1, 1, 1)

. Latin indices run from one to three, Greek ones from zero to three; the latter are raised and lowered by the Minkowski metric. The wave 4-vector is k_μ=(−ω,k)

k_{μ} = (- ω, k)

, k=k(ω)k₀

k = k (ω) k_{0}

, where k₀

k_{0}

is the unit wave vector and k(ω)

k (ω)

the wave number depending on the permeability tensor. In the aether frame, the permeability tensor is isotropic,

equation1.2

g^{00} = - ε (ω), g^{i k} = δ^{i k} / μ (ω), g^{0 k} = 0,

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where ε and μ are positive frequency-dependent permeabilities.

In Section 2, we discuss neutrino oscillations based on a massive Dirac equation (1.1) coupled by a permeability tensor (1.2) to the aether. In Section 3, we demonstrate that the interference phase can be generated by dispersive permeabilities (1.2) without a mass term in the wave equation. In Section 4, we study gauge and Proca fields in a permeable spacetime, in analogy to spinor fields. Each gauge component admits a permeability tensor which produces a massive dispersion relation without breaking the gauge invariance. In Section 5, we explain how permeability tensors generate effective mass in the propagators of massless gauge and Dirac fields. In Section 6, we study Dirac neutrinos in moving inertial frames coupled to an anisotropic permeability tensor and discuss Michelson–Morley neutrino experiments. In Section 7, we present our conclusions.

2. Neutrino mixing in a dispersive spacetime

We start with the dispersion relations [10]

equation2.1

k_{(j)} (ω) = μ_{(j)} (ω) \sqrt{ε_{(j)}^{2} (ω) ω^{2} - m_{(j)}^{2}},

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derived by squaring the Dirac equation, where j labels three mass eigenstates ψ_j

ψ_{j}

, each characterized by permeabilities ε_(j)(ω)

ε_{(j)} (ω)

, μ_(j)(ω)

μ_{(j)} (ω)

, and a neutrino mass m_(j)

m_{(j)}

; the k_(j)(ω)

k_{(j)} (ω)

are the respective wave numbers. Each eigenstate ψ_j

ψ_{j}

satisfies a Dirac equation with mass m_(j)

m_{(j)}

and a permeability tensor

g_{(j)}^{μ ν} (ω)

defined by dispersive permeabilities ε_(j)(ω)

ε_{(j)} (ω)

and μ_(j)(ω)

μ_{(j)} (ω)

. We use the shortcuts k_(j)=k_(j)(ω_j)

k_{(j)} = k_{(j)} (ω_{j})

, ε_(j)=ε_(j)(ω_j)

ε_{(j)} = ε_{(j)} (ω_{j})

and μ_(j)=μ_(j)(ω_j)

μ_{(j)} = μ_{(j)} (ω_{j})

, where ω_j

ω_{j}

is the frequency of the eigenstate plane wave ψ_j

ψ_{j}

with wave 4-vector

k_{μ}^{j} = (- ω_{j}, k_{(j)} k_{0})

. The flavor transition amplitude reads [1]

equation2.2

A (ν_{l} \to ν_{l^{'}}) = \sum_{j = 1}^{3} U_{l j} D_{j} U_{j l^{'}}^{†},

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where l,l^′=e,μ,τ

l, l^{'} = e, μ, τ

are the flavor indices of Dirac neutrinos and j is the eigenstate index. U_lj

U_{l j}

is a unitary matrix parametrized with Eulerian mixing angles and a CP violating phase. The D_j

D_{j}

denote phase factors of the plane waves ψ_j

ψ_{j}

equation2.3

D_j=exp(i(k_(j)L−ω_jT)),

D_{j} = \exp (i (k_{(j)} L - ω_{j} T)),

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L = k_{0} (x_{abs} - x_{em}), T = t_{abs} - t_{em},

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where the subscripts refer to emission and absorption. The neutrino wave vector is k_(j)=k_(j)k₀

k_{(j)} = k_{(j)} k_{0}

, L is the path length between source and detector, and T the time of flight.

The transition probability |A(ν_l→ν_l^′)|² ${| A (ν_{l} \to ν_{l^{'}}) |}^{2}$ depends on the phase factors $D_{i} D_{j}^{⁎} = \exp (- i δ φ_{i j})$ , defined by the phase difference

equation2.4

δ φ_{i j} = δ ω_{i j} T - δ k_{i j} L, δ ω_{i j} = ω_{i} - ω_{j},

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δk_ij=k_(i)(ω_i)−k_(j)(ω_j).

δ k_{i j} = k_{(i)} (ω_{i}) - k_{(j)} (ω_{j}) .

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The increments

equation2.5

Δ m_{i j}^{2} = m_{(i)}^{2} - m_{(j)}^{2},

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Δε_ij(ω)=ε_(i)(ω)−ε_(j)(ω),

Δ ε_{i j} (ω) = ε_{(i)} (ω) - ε_{(j)} (ω),

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Δμ_ij(ω)=μ_(i)(ω)−μ_(j)(ω),

Δ μ_{i j} (ω) = μ_{(i)} (ω) - μ_{(j)} (ω),

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as well as δω_ij

δ ω_{i j}

in (2.4) are treated as small variations, and we expand δk_ij

δ k_{i j}

in linear order,

equation2.6

δ k_{i j} \sim \frac{\partial k_{j}}{\partial ω_{j}} δ ω_{i j} + \frac{\partial k_{j}}{\partial μ_{j}} Δ μ_{i j} (ω_{j}) + \frac{\partial k_{j}}{\partial ε_{j}} Δ ε_{i j} (ω_{j}) + \frac{\partial k_{j}}{\partial m_{j}^{2}} Δ m_{i j}^{2} .

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The k_(j)(ω_j)

k_{(j)} (ω_{j})

are the eigenstate wave numbers (2.1), whose frequency derivative is the reciprocal group velocity, ∂k_(j)/∂ω_j=1/υ_gr(j)

\partial k_{(j)} / \partial ω_{j} = 1 / υ_{gr (j)}

[11]. We thus find

equation2.7

δ k_{i j} \sim \frac{1}{υ_{gr (j)}} δ ω_{i j} + \frac{k_{(j)}}{μ_{(j)}} Δ μ_{i j} (ω_{j}) + μ_{(j)}^{2} ε_{(j)} \frac{ω_{j}^{2}}{k_{(j)}} Δ ε_{i j} (ω_{j}) - \frac{1}{2} \frac{μ_{(j)}^{2}}{k_{(j)}} Δ m_{i j}^{2} .

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In the phase difference δφ_ij

δ φ_{i j}

in (2.4), we put T=L/υ_gr(j)

T = L / υ_{gr (j)}

and substitute δk_ij

δ k_{i j}

equation2.8

\frac{δ φ_{i j}}{L} \sim \frac{1}{2} \frac{μ_{(j)}^{2}}{k_{(j)}} Δ m_{i j}^{2} - \frac{k_{(j)}}{μ_{(j)}} Δ μ_{i j} (ω_{j}) - \frac{μ_{(j)}^{2} ε_{(j)} ω_{j}^{2}}{k_{(j)}} Δ ε_{i j} (ω_{j}) .

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If the time of flight T is defined by υ_gr(i)

υ_{gr (i)}

instead of υ_gr(j)

υ_{gr (j)}

, this does not affect δφ_ij

δ φ_{i j}

in linear order, as these two velocities only differ by delta increments. For the same reason, we can drop the index j from ω. Similarly, if we replace the eigenstate index j by i in the three ratios on the right-hand side of (2.8), this does not affect the indicated linear order.

We split the eigenstate permeabilities and mass-squares in (2.1) as

equation2.9

m_{(j)}^{2} = m^{2} + δ m_{(j)}^{2},

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ε_(j)(ω)=ε(ω)+δε_(j)(ω),

ε_{(j)} (ω) = ε (ω) + δ ε_{(j)} (ω),

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μ_(j)(ω)=μ(ω)+δμ_(j)(ω),

μ_{(j)} (ω) = μ (ω) + δ μ_{(j)} (ω),

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where the increments

δ m_{(j)}^{2}

, δε_(j)(ω)

δ ε_{(j)} (ω)

and δμ_(j)(ω)

δ μ_{(j)} (ω)

are small deviations from base values m²

m^{2}

(which can be zero) and ε(ω)

ε (ω)

, μ(ω)

μ (ω)

(both close to 1, see after (3.3)). In the ratios of Eq. (2.8), we replace the permeabilities and the mass-squares in the wave numbers by their base values, since the expansion is linear in the delta increments. We thus arrive at the interference phase

equation2.10

δ φ_{i j} (ω) \sim (\frac{μ^{2}}{2 k} Δ m_{i j}^{2} - \frac{k}{μ} Δ μ_{i j} (ω) - \frac{μ^{2} ε ω^{2}}{k} Δ ε_{i j} (ω)) L,

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where the wave number

k = μ (ω) \sqrt{ε^{2} (ω) ω^{2} - m^{2}}

is independent of the eigenstate index.

3. Mass eigenstates generated by permeability tensors

If the mass eigenstates admit the same permeabilities, ε_(j)(ω)=ε(ω) $ε_{(j)} (ω) = ε (ω)$ , μ_(j)(ω)=μ(ω) $μ_{(j)} (ω) = μ (ω)$ , cf. (2.9), we can put Δμ_ij=Δε_ij=0 $Δ μ_{i j} = Δ ε_{i j} = 0$ in the interference phase (2.10). Neglecting the mass-square m² $m^{2}$ in the wave number k, cf. (2.9) and after (2.10), we find

equation3.1

δ φ_{i j} (ω) \sim \frac{1}{2} \frac{μ (ω)}{ε (ω) ω} Δ m_{i j}^{2} L .

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Alternatively, we may assume that the mass eigenstates ψ_j

ψ_{j}

admit the same mass-square

m_{(j)}^{2} = m^{2}

, so that

Δ m_{i j}^{2} = 0

. In this case, the interference phase (2.10) stems from the permeability increments (2.5). We put m²=0

m^{2} = 0

in (2.9), so that k=μ(ω)ε(ω)ω

k = μ (ω) ε (ω) ω

and

equation3.2

δφ_ij(ω)∼−(ε(ω)Δμ_ij+μ(ω)Δε_ij)ωL,

δ φ_{i j} (ω) \sim - (ε (ω) Δ μ_{i j} + μ (ω) Δ ε_{i j}) ω L,

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where Δμ_ij(ω)=δμ_(i)(ω)−δμ_(j)(ω)

Δ μ_{i j} (ω) = δ μ_{(i)} (ω) - δ μ_{(j)} (ω)

and analogously Δε_ij(ω)

Δ ε_{i j} (ω)

. The interference phase (3.1) is recovered from (3.2) by specifying the increments δε_(j)(ω)

δ ε_{(j)} (ω)

and δμ_(j)(ω)

δ μ_{(j)} (ω)

of the eigenstate permeabilities (2.9) as

equation3.3

δ ε_{(j)} (ω) = - \frac{a (ω)}{2} \frac{1}{ε (ω)} \frac{m_{(j)}^{2}}{ω^{2}},

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δ μ_{(j)} (ω) = \frac{a (ω) - 1}{2} \frac{μ (ω)}{ε^{2} (ω)} \frac{m_{(j)}^{2}}{ω^{2}} .

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Here, a(ω)

a (ω)

is an arbitrary real function that drops out in the phase difference (3.2); we may conveniently choose a=0,1

a = 0, 1

, or a=μ/(ε+μ)≈1/2

a = μ / (ε + μ) \approx 1 / 2

. Thus we have shown that the interference phase (3.1) is reproduced by the eigenstate permeabilities ε_(j)(ω)

ε_{(j)} (ω)

and μ_(j)(ω)

μ_{(j)} (ω)

defined in (2.9) and (3.3), without inserting mass terms m_(j)ψ_j

m_{(j)} ψ_{j}

in the Dirac equations of the eigenstate plane waves. The refractive indices of the eigenstates ψ_j

ψ_{j}

read, in leading order in

m_{(j)}^{2} / ω^{2}

, cf. (2.9) and (3.3),

equation3.4

n_{r (j)} (ω) = μ_{(j)} (ω) ε_{(j)} (ω) \sim ε (ω) μ (ω) - \frac{1}{2} \frac{μ (ω)}{ε (ω)} \frac{m_{(j)}^{2}}{ω^{2}},

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and the wave numbers k_(j)(ω)=n_r(j)(ω)ω

k_{(j)} (ω) = n_{r (j)} (ω) ω

admit massive dispersion relations

k_{(j)}^{2} (ω) \sim μ^{2} (ε^{2} ω^{2} - m_{(j)}^{2})

with an effective neutrino mass m_(j)

m_{(j)}

generated by the permeability increments (3.3).

A bound on the neutrino refractive index n_r∼1/υ_gr∼k(ω)/ω∼μ(ω)ε(ω) $n_{r} \sim 1 / υ_{gr} \sim k (ω) / ω \sim μ (ω) ε (ω)$ [11] at ω∼17 GeV $ω \sim 17 GeV$ is |1−n_r|<3×10⁻⁶ $| 1 - n_{r} | < 3 \times 10^{- 6}$ , based on recent measurements of the neutrino speed by the OPERA, BOREXINO, LVD and ICARUS Collaborations [6], [7], [8] and [9]. We use the n_r(ω) $n_{r} (ω)$ estimate also for the factors μ(ω) $μ (ω)$ and ε(ω) $ε (ω)$ . The neutrino flux from supernova SN1987A gives the bound |υ_gr−1|<2×10⁻⁹ $| υ_{gr} - 1 | < 2 \times 10^{- 9}$ at 10 MeV [22], so that we put μ(ω)≈ε(ω)≈1 $μ (ω) \approx ε (ω) \approx 1$ in the low MeV region. An upper bound on the eigenstate masses m_(j) $m_{(j)}$ from tritium β decay is 2 eV [23], assuming vacuum permeabilities in the eV range. The neutrino frequency ω is in the low MeV region (for reactor and solar ν^′ $ν^{'}$ s) or low GeV region (accelerator, atmospheric ν^′ $ν^{'}$ s). The path length L is of order 10³ km $10^{3} km$ and 10⁴ km $10^{4} km$ for accelerator and atmospheric ν^′ $ν^{'}$ s. The differences $Δ m_{i j}^{2}$ of the eigenstate mass-squares are of order 10⁻⁴ eV² $10^{- 4} {eV}^{2}$ or 10⁻³ eV² $10^{- 3} {eV}^{2}$ [1] and [5]. To generate flavor oscillations, a phase difference |δφ_ij|⩾1 $| δ φ_{i j} | ⩾ 1$ is required for substantial interference in the squared amplitudes (2.2). As for the neutrino speed determined by the refractive index n_r(j) $n_{r (j)}$ in (3.4), we can ignore terms depending on the tiny (compared to O(10⁻⁶) $O (10^{- 6})$ ) ratios $m_{(j)}^{2} / ω^{2}$ , at least at length scales relevant for terrestrial experiments and solar ν^′ $ν^{'}$ s. Accordingly, n_r∼μ(ω)ε(ω) $n_{r} \sim μ (ω) ε (ω)$ is independent of the eigenstate index, so that all neutrino flavors admit the same group velocity υ_gr=1/(ωn_r(ω))^′∼1/n_r $υ_{gr} = 1 / (ω n_{r} {(ω))}^{'} \sim 1 / n_{r}$ .

4. Gauge and Proca fields in the aether: Massive dispersion relations preserving gauge invariance

The Lagrangian of a Proca field with negative mass-square reads

equation4.1

L = - \frac{1}{4} F_{α β} g_{F}^{α β μ ν} F_{μ ν} + \frac{1}{2} m_{t}^{2} A_{μ} g_{A}^{μ ν} A_{ν} - \frac{1}{2 ξ} {(g_{A}^{μ ν} A_{μ, ν})}^{2} + A_{μ} g_{J}^{μ ν} j_{ν},

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where

η^{α μ} η^{β ν} \to g_{F}^{α β μ ν}

η^{μ ν} \to g_{A}^{μ ν}

, and

η^{μ ν} \to g_{J}^{μ ν}

are permeability tensors replacing the Minkowski metric η^μν=diag(−1,1,1,1)

η^{μ ν} = diag (- 1, 1, 1, 1)

in the vacuum Lagrangian [24] and [25]. The signs in (4.1) are chosen in a way that

m_{t}^{2} > 0

is the negative mass-square of the tachyonic Proca field A_μ

A_{μ}

, and F_μν=A_ν,μ−A_μ,ν

F_{μ ν} = A_{ν, μ} - A_{μ, ν}

is the field tensor. A gauge-fixing term ∝1/ξ

\propto 1 / ξ

is included as Lagrange multiplier. Massive Proca fields are obtained by replacing

m_{t}^{2} \to - m^{2}

. The 3D field strengths are E_i=F_i0

E_{i} = F_{i 0}

, B^k=ε^kijF_ij/2

B^{k} = ε^{k i j} F_{i j} / 2

, and inversely F_ij=ε_ijkB^k

F_{i j} = ε_{i j k} B^{k}

. Thus, E_i=A_0,i−A_i,0

E_{i} = A_{0, i} - A_{i, 0}

, B^k=ε^kijA_j,i

B^{k} = ε^{k i j} A_{j, i}

, where ε_ijk

ε_{i j k}

is the totally antisymmetric tensor. The field is coupled to a current j_μ=(−ρ,j)

j_{μ} = (- ρ, j)

. The permeability tensors

g_{F}^{α β μ ν} (ω)

and

g_{A, J}^{μ ν} (ω)

are real and frequency dependent. The

g_{A, J}^{μ ν}

are symmetric, and

g_{F}^{α β μ ν}

is antisymmetric in the first as well as second index pair, and symmetric regarding the interchange αβ↔μν

α β \leftrightarrow μ ν

of the index pairs. We define the inductive potential, field tensor and current as

equation4.2

C^{μ} = g_{A}^{μ ν} A_{ν}, H^{α β} = g_{F}^{α β μ ν} F_{μ ν}, j_{Ω}^{μ} = g_{J}^{μ ν} j_{ν},

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and write

j_{Ω}^{μ} = (ρ_{Ω}, j_{Ω})

, so that j_Ω0=−ρ_Ω

j_{Ω 0} = - ρ_{Ω}

. Greek indices are raised and lowered with the Minkowski metric. The Lagrangian (4.1) can thus be written as

equation4.3

L = - \frac{1}{4} F_{α β} H^{α β} + \frac{1}{2} m_{t}^{2} A_{μ} C^{μ} - \frac{1}{2 ξ} {(C_{, μ}^{μ})}^{2} + A_{μ} j_{Ω}^{μ} .

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The 3D inductions are Dⁱ=H⁰ⁱ

D^{i} = H^{0 i}

, H_i=ε_imnH^mn/2

H_{i} = ε_{i m n} H^{m n} / 2

, and inversely H^mn=H_jε^jmn

H^{m n} = H_{j} ε^{j m n}

, so that F_αβH^αβ=2(BH–ED)

F_{α β} H^{α β} = 2 (BH – ED)

. The constitutive equations relating the 3D inductions to the field strengths read

equation4.4

D^{i} = ε^{i j} E_{j} + κ_{k}^{i} B^{k}, H_{i} = μ_{i j} B^{j} - κ_{i}^{j} E_{j},

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where the permeabilities ε^ij(ω)

ε^{i j} (ω)

and μ_ij(ω)

μ_{i j} (ω)

are real symmetric 3-tensors, and

κ_{k}^{i} (ω)

is a mixed tensor. We need not assume different tensors

κ_{k}^{i}

in the two relations (4.4), as the Lagrangian would only depend on their sum via F_αβH^αβ

F_{α β} H^{α β}

. The permeability tensor

g_{F}^{α β μ ν}

in (4.1) is defined by these 3-tensors,

equation4.5

g_{F}^{0 i 0 j} = - \frac{1}{2} ε^{i j}, g_{F}^{0 i a b} = \frac{1}{2} κ_{k}^{i} ε^{k a b},

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g_{F}^{a b c d} = \frac{1}{2} ε^{a b i} μ_{i k} ε^{k c d}, g_{F}^{m n k 0} = - \frac{1}{2} κ_{i}^{k} ε^{i m n} .

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All other components of

g_{F}^{α β μ ν}

follow from the mentioned symmetries or vanish. Inversely,

equation4.6

ε^{i j} = - 2 g_{F}^{0 i 0 j}, μ_{i k} = \frac{1}{2} ε_{i a b} g_{F}^{a b m n} ε_{m n k},

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κ_{j}^{i} = g_{F}^{0 i m n} ε_{m n j} .

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In the isotropic aether frame, we can put

κ_{i}^{j} = 0

, cf. after (4.11), but in moving frames the tensor

κ_{i}^{j}

does not vanish, cf. after (5.4).

Euler variation of Lagrangian (4.3) gives the field equations and gauge condition

equation4.7

H_{, ν}^{μ ν} - m_{t}^{2} C^{μ} - \frac{1}{ξ} g_{A}^{μ ν} C_{, κ, ν}^{κ} = j_{Ω}^{μ},

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equation4.8

\frac{1}{ξ} g_{A}^{μ ν} C_{, κ, μ, ν}^{κ} + m_{t}^{2} C_{, κ}^{κ} = - j_{Ω, μ}^{μ} .

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For instance, the Lorentz condition

C_{, κ}^{κ} = 0

follows from current conservation

j_{Ω, μ}^{μ} = 0

if the mass-square does not vanish and the gauge-fixing term in the Lagrangian is dropped, ξ=∞

ξ = \infty

. Eqs. (4.7) are equivalent to

equation4.9

D_{, i}^{i} + m_{t}^{2} C_{0} - \frac{1}{ξ} g_{A}^{0 ν} C_{, κ, ν}^{κ} = - j_{Ω 0},

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ε^{i k n} H_{n, k} - D_{, 0}^{i} - m_{t}^{2} C^{i} - \frac{1}{ξ} g_{A}^{i ν} C_{, κ, ν}^{κ} = j_{Ω}^{i},

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where j_Ω0=−ρ_Ω

j_{Ω 0} = - ρ_{Ω}

, and the constitutive relations (4.4) apply. In the massless case

m_{t}^{2} = 0

, the gauge-invariant Maxwell equations in a permeable spacetime are recovered by imposing the Lorentz condition

C_{, κ}^{κ} = 0

on the vector potential, also see (4.11).

We consider isotropic permeability tensors, cf. (4.2), (4.4) and (4.5),

equation4.10

ε^{i k} = ε δ^{i k}, μ_{i k} = (1 / μ) δ_{i k}, κ_{i}^{j} = κ δ_{i}^{j},

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g_{A}^{00} = - ε_{0}, g_{A}^{i j} = δ^{i j} / μ_{0}, g_{A}^{k 0} = 0,

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g_{J}^{00} = - Ω_{0}, g_{J}^{m n} = δ^{m n} / Ω, g_{J}^{k 0} = 0,

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with permeabilities (ε,μ)

(ε, μ)

, (ε₀,μ₀)

(ε_{0}, μ_{0})

and (Ω₀,Ω)

(Ω_{0}, Ω)

depending on the frequency ω of the wave modes in the aether frame [11]. The tensor

g_{J}^{α β}

couples the current to the wave modes, cf. (4.1) and (4.2), and amounts to a varying coupling constant if Ω₀(ω)

Ω_{0} (ω)

coincides with 1/Ω(ω)

1 / Ω (ω)

[20]. We note H^k=ε^kijA_j,i/μ

H^{k} = ε^{k i j} A_{j, i} / μ

and D_k=ε(A_0,k−A_k,0)

D_{k} = ε (A_{0, k} - A_{k, 0})

, cf. after (4.1) and (4.4). The isotropic field equations for the vector potential read, cf. (4.9),

equation4.11

A_{0, k, k} - A_{k, k, 0} + m_{t}^{2} \frac{ε_{0}}{ε} A_{0} + \frac{1}{ξ} \frac{ε_{0}}{ε} C_{, κ, 0}^{κ} = - \frac{1}{ε} j_{Ω 0},

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A_{i, k, k} - A_{k, k, i} + ε μ (A_{0, 0, i} - A_{i, 0, 0}) + m_{t}^{2} \frac{μ}{μ_{0}} A_{i} + \frac{1}{ξ} \frac{μ}{μ_{0}} C_{, κ, i}^{κ} = - μ j_{Ω i},

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where

C_{, κ}^{κ} = - ε_{0} A_{0, 0} + A_{k, k} / μ_{0}

. The isotropic tensor

κ_{i}^{j} = κ δ_{i}^{j}

in the constitutive relations (4.4) drops out in the field equations, since the κEB term in the Lagrangian (see after (4.3)) is a divergence, 2E_kB^k=(ε^αβμνA_βA_μ,ν)_,α

2 E_{k} B^{k} = {(ε^{α β μ ν} A_{β} A_{μ, ν})}_{, α}

. In [19] and [20], we studied the field Eqs. (4.11) in the limit ξ=∞

ξ = \infty

. In the following, we consider the opposite limit,

m_{t}^{2} = 0

with finite ξ. That is, we drop the mass term in Lagrangian (4.1) and generate a positive or negative mass-square in the dispersion relation by way of the permeability tensors (4.10).

The isotropic field equations (4.11) (with $m_{t}^{2} = 0, j_{Ω}^{μ} = 0$ ) admit the plane-wave solutions

equation4.12

A_{0} (x, t) = - \frac{ω}{k_{L}} {\hat{A} k}_{0} e^{i (k_{L} (ω) k_{0} x - ω t)} + c.c.,

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A (x, t) = {\hat{A}}_{⊥} e^{i (k_{T} (ω) k_{0} x - ω t)} + ({\hat{A} k}_{0}) k_{0} e^{i (k_{L} (ω) k_{0} x - ω t)} + c.c.,

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where we have split the field into a transversal and longitudinal component,

\hat{A} = {\hat{A}}_{⊥} + ({\hat{A} k}_{0}) k_{0}, {\hat{A}}_{⊥} k_{0} = 0

, with wave number

k_{T}^{2} = ε μ ω^{2}

and

k_{L}^{2} = ε_{0} μ_{0} ω^{2}

, respectively. The gauge-fixing constant ξ does not explicitly show in (4.12). If the gauge-fixing term is dropped by putting ξ=∞

ξ = \infty

, the general plane-wave solution of frequency ω is still (4.12), but with k_L(ω)

k_{L} (ω)

unspecified. A finite ξ gives a well-defined wave number to the longitudinal component

\propto {\hat{A} k}_{0}

, which is a gauge transformation,

A_{μ} = A_{μ}^{⊥} + λ_{, μ}

λ = {\hat{A} k}_{0} e^{i (k_{L} k_{0} x - ω t)} / (i k_{L}) + c.c

The permeability tensor $g_{A}^{μ ν}$ defines the gauge-fixing term in Lagrangian (4.3) $(m_{t}^{2} = 0)$ . We identify ε₀=ε $ε_{0} = ε$ and μ₀=μ $μ_{0} = μ$ , so that $g_{A}^{μ ν}$ in (4.10) reads $g_{A}^{00} = - ε$ , $g_{A}^{i j} = δ^{i j} / μ$ , $g_{A}^{k 0} = 0$ . The transversal and longitudinal components of the vector potential (4.12) thus admit the same wave number, $k^{2} = k_{T, L}^{2} = ε μ ω^{2}$ , where ε(ω) $ε (ω)$ and μ(ω) $μ (ω)$ are the permeabilities defining $g_{F}^{κ λ μ ν}$ , cf. (4.5) and (4.10). A massive dispersion relation k²=ω²−m² $k^{2} = ω^{2} - m^{2}$ is induced by permeabilities satisfying εμ=1−m²/ω² $ε μ = 1 - m^{2} / ω^{2}$ , without a mass term in the Lagrangian, and longitudinal field components remain gauge transformations. If we impose the Lorentz condition $C_{, κ}^{κ} = 0$ , the gauge-fixing term drops out in the field equations (4.11) (with $m_{t}^{2} = 0$ ), and they become gauge invariant despite of the massive dispersion relation induced by the permeabilities. [A negative mass-square in the dispersion relation requires permeabilities related by $ε μ = 1 + m_{t}^{2} / ω^{2}$ . A tachyonic dispersion relation preserving the hermiticity of the Dirac Hamiltonian is obtained by replacing $m_{(j)}^{2} \to - m_{t (j)}^{2}$ in (5.14).] As for electromagnetic fields, we use vacuum permeabilities, ε₀=ε=1 $ε_{0} = ε = 1$ and μ₀=μ=1 $μ_{0} = μ = 1$ , defining the constant speed in the Lorentz boosts, so that the electromagnetic Michelson–Morley isotropy is preserved in moving frames.

5. Effective mass-squares in propagators of dispersive gauge and Dirac fields

The foregoing admits generalization to non-Abelian gauge fields $A_{μ}^{α}$ , each component being coupled to the aether by permeabilities ε_(α)(ω) $ε_{(α)} (ω)$ and μ_(α)(ω) $μ_{(α)} (ω)$ , α=1,…,N $α = 1, \dots, N$ . We consider the Lagrangian, cf. (4.3),

equation5.1

L = - \frac{1}{4} F_{κ λ}^{α} H_{α}^{κ λ} - \frac{1}{2 ξ} {(C_{α, μ}^{μ})}^{2} + A_{μ}^{α} j_{Ω α}^{μ},

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with field tensor

F_{μ ν}^{α} = A_{ν, μ}^{α} - A_{μ, ν}^{α} - g f_{β γ}^{α} A_{μ}^{β} A_{ν}^{γ}

, inductions

H_{α}^{κ λ} = g_{F (α)}^{κ λ μ ν} F_{μ ν}^{α}

C_{α}^{μ} = g_{A (α)}^{μ ν} A_{ν}^{α}

, and inductive current

j_{Ω α}^{μ} = g_{J (α)}^{μ ν} j_{ν}^{α}

. We write the gauge index α of the permeability tensors

g_{F (α)}^{κ λ μ ν}

g_{A (α)}^{μ ν}

and

g_{J (α)}^{μ ν}

in parentheses, to indicate that it is not a summation index unless in a product where α appears twice without parentheses. When calculating the propagator, it suffices to use the quadratic component of the Lagrangian and to put the coupling constant g in the field tensor to zero, so that the linearized field equations are independent of the structure constants

f_{β γ}^{α}

H_{α, ν}^{μ ν} - C_{α, κ, ν}^{κ} g_{A (α)}^{ν μ} / ξ = j_{Ω α}^{μ}

, cf. (4.7). The tensors

g_{A (α)}^{μ ν}

and

g_{J (α)}^{μ ν}

are isotropic in the aether frame, cf. (4.8) and (4.10) and after (4.12):

equation5.2

g_{A (α)}^{00} = - ε_{(α)} (ω), g_{A (α)}^{i j} = δ^{i j} / μ_{(α)} (ω), g_{A (α)}^{0 i} = 0,

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g_{J (α)}^{00} = - Ω_{0 (α)} (ω), g_{J (α)}^{i j} = δ^{i j} / Ω_{(α)} (ω), g_{J (α)}^{0 i} = 0 .

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The isotropic permeability tensor

g_{F (α)}^{κ λ μ ν}

reads, cf. (4.5) and (4.10),

equation5.3

g_{F (α)}^{0 i 0 j} = - \frac{1}{2} ε_{(α)} δ^{i j}, g_{F (α)}^{0 i m n} = \frac{1}{2} κ_{(α)} ε^{i m n},

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g_{F (α)}^{k l m n} = \frac{δ^{k m} δ^{l n} - δ^{k n} δ^{l m}}{2 μ_{(α)} (ω)},

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with all other components following from symmetry properties, cf. after (4.1). Without loss of generality, we can put κ_(α)(ω)=0

κ_{(α)} (ω) = 0

, cf. after (4.11), and reduce

g_{F (α)}^{κ λ μ ν}

to a symmetric tensor

g_{(α)}^{μ ν} = μ_{(α)}^{1 / 2} g_{A (α)}^{μ ν}

equation5.4

g_{F (α)}^{κ λ μ ν} = \frac{1}{2} (g_{(α)}^{κ μ} g_{(α)}^{λ ν} - g_{(α)}^{κ ν} g_{(α)}^{λ μ}), H_{α}^{κ λ} = g_{(α)}^{κ μ} g_{(α)}^{λ ν} F_{μ ν}^{α} .

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The refractive indices

n_{r (α)} = \sqrt{ε_{(α)} μ_{(α)}} = \sqrt{1 - m_{(α)}^{2} / ω^{2}}

give rise to massive dispersion relations k_(α)=ωn_r(α)

k_{(α)} = ω n_{r (α)}

with an effective gauge mass m_(α)

m_{(α)}

for each gauge component

A_{μ}^{α} = (A_{0}^{α}, A^{α})

, cf. (4.12). The tensors

g_{(α)}^{μ ν}

transform contravariantly under Lorentz boosts,

g_{(α)}^{' μ ν} = g_{(α)}^{κ λ} Λ_{κ}^{(- 1) μ} Λ_{λ}^{(- 1) ν}

cf. [11] and Section 6.

We substitute the plane waves $A_{μ}^{α} = {\hat{A}}_{μ}^{α} e^{i (kx - ω t)} + c.c.$ and $j_{Ω α}^{μ} = {\hat{j}}_{Ω α}^{μ} e^{i (kx - ω t)} + c.c.$ into the linearized isotropic Lagrangian (5.1),

equation5.5

L = - \frac{1}{4} \frac{1}{μ_{(α)}} F_{m l}^{α} F_{m l}^{α} + \frac{1}{2} ε_{(α)} F_{0 n}^{α} F_{0 n}^{α} - \frac{1}{2 ξ} {(A_{i, i}^{α} / μ_{(α)} - ε_{(α)} A_{0, 0}^{α})}^{2} + A_{μ}^{α} j_{Ω α}^{μ},

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and perform a time average over a period to find

equation5.6

L = - {\hat{A}}_{μ}^{α} {\hat{A}}_{ν}^{β ⁎} {\hat{G}}_{α β}^{(- 1) μ ν} + {\hat{A}}_{μ}^{α} {\hat{j}}_{Ω α}^{μ ⁎} + {\hat{A}}_{μ}^{α ⁎} {\hat{j}}_{Ω α}^{μ},

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where

{\hat{G}}_{α β}^{(- 1) μ ν}

denotes the Gaussian kernel

equation5.7

{\hat{G}}_{α β}^{(- 1) 00} = - δ_{α β} ε_{(α)} (k^{2} - \frac{ε_{(α)}}{ξ} ω^{2}),

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{\hat{G}}_{α β}^{(- 1) m 0} = δ_{α β} ε_{(α)} (\frac{1}{ξ μ_{(α)}} - 1) ω k_{m},

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{\hat{G}}_{α β}^{(- 1) m n} = \frac{δ_{α β}}{μ_{(α)}} [(k^{2} - ε_{(α)} μ_{(α)} ω^{2}) δ_{m n} + (\frac{1}{ξ μ_{(α)}} - 1) k_{m} k_{n}] .

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Euler variation of L with respect to

{\hat{A}}_{μ}^{α ⁎}

gives

{\hat{G}}_{α β}^{(- 1) μ ν} {\hat{A}}_{μ}^{α} = {\hat{j}}_{Ω β}^{ν}

. The inverse kernel is the propagator

{\hat{G}}_{μ ν}^{α β}

equation5.8

{\hat{G}}_{00}^{α β} = - \frac{δ^{α β}}{ε_{(α)}} \frac{k^{2} - ξ μ_{(α)}^{2} ε_{(α)} ω^{2}}{{(k^{2} - μ_{(α)} ε_{(α)} ω^{2})}^{2}},

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{\hat{G}}_{m 0}^{α β} = δ^{α β} \frac{μ_{(α)} (1 - ξ μ_{(α)}) ω k_{m}}{{(k^{2} - μ_{(α)} ε_{(α)} ω^{2})}^{2}},

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{\hat{G}}_{m n}^{α β} = \frac{δ^{α β} μ_{(α)}}{k^{2} - ε_{(α)} μ_{(α)} ω^{2}} (δ_{m n} + \frac{(ξ μ_{(α)} - 1) k_{m} k_{n}}{k^{2} - ε_{(α)} μ_{(α)} ω^{2}}) .

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At ξ=0

ξ = 0

, this propagator satisfies the Lorentz condition

g_{A (α)}^{μ ν} k_{μ} {\hat{G}}_{ν λ}^{α β} = 0

, where k_μ=(−ω,k)

k_{μ} = (- ω, k)

. A frequency-dependent gauge parameter is also admissible, which can even be different for each gauge component α; for instance, we may substitute ξ→ξ_(α)=1/μ_(α)

ξ \to ξ_{(α)} = 1 / μ_{(α)}

, which diagonalizes the propagator. Observable gauge-invariant quantities such as the linear inductions

H_{α}^{k} = ε^{k i j} A_{j, i}^{α} / μ_{(α)}

and

D_{k}^{α} = ε_{(α)} (A_{0, k}^{α} - A_{k, 0}^{α})

obtained from the solution

{\hat{A}}_{μ}^{α} = {\hat{G}}_{μ ν}^{α β} {\hat{j}}_{Ω β}^{ν}

are independent of the gauge parameter ξ, and the same holds true for the field strengths

E_{k}^{α} = D_{k}^{α} / ε_{(α)}

and

B_{α}^{k} = μ_{(α)} H_{α}^{k}

equation5.9

E_{n}^{α} = {\hat{E}}_{n}^{α} e^{i (kx - ω t)} + c.c., {\hat{E}}_{n}^{α} = i \frac{ω μ_{(α)} ε_{(α)} {\hat{j}}_{Ω α n} - k_{n} {\hat{j}}_{Ω α}^{0}}{ε_{(α)} (k^{2} - μ_{(α)} ε_{(α)} ω^{2})},

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B_{α}^{n} = {\hat{B}}_{α}^{n} e^{i (kx - ω t)} + c.c., {\hat{B}}_{α}^{n} = \frac{i μ_{(α)} ε^{n i j} k_{i} {\hat{j}}_{Ω α j}}{k^{2} - ε_{(α)} μ_{(α)} ω^{2}} .

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A more compact representation of kernel (5.7) and propagator (5.8) is

equation5.10

{\hat{G}}_{α β}^{(- 1) μ ν} = δ_{α β} [g_{(α)}^{μ ν} g_{(α)}^{κ λ} k_{κ} k_{λ} + (\frac{1}{ξ μ_{(α)}} - 1) g_{(α)}^{μ κ} g_{(α)}^{ν λ} k_{κ} k_{λ}],

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equation5.11

{\hat{G}}_{μ ν}^{α β} = \frac{δ^{α β}}{g_{(α)}^{κ λ} k_{κ} k_{λ}} [g_{(α) μ ν}^{(- 1)} + (ξ μ_{(α)} - 1) \frac{k_{μ} k_{ν}}{g_{(α)}^{κ λ} k_{κ} k_{λ}}],

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where

g_{(α) μ ν}^{(- 1)}

denotes the inverse of the diagonal matrix

g_{(α)}^{μ ν} = μ_{(α)}^{1 / 2} g_{A (α)}^{μ ν}

, cf. (5.2) and (5.4). We can replace

g_{A (α)}^{μ ν} \to g_{(α)}^{μ ν}

in the gauge-fixing term of Lagrangian (5.1), so that the inductive field is

C_{α}^{μ} = g_{(α)}^{μ ν} A_{ν}^{α}

. This amounts to a conformal rescaling ξ→ξ/μ_(α)

ξ \to ξ / μ_{(α)}

of the gauge parameter in propagator (5.11), which becomes manifestly covariant in this gauge,

g_{(α)}^{μ ν} C_{α, κ, μ, ν}^{κ} + ξ j_{Ω α, μ}^{μ} = 0

, cf. (4.8) and after (5.8).

We specify the permeabilities as $ε_{(α)} μ_{(α)} \sim 1 - (m_{(α)}^{2} - i {\tilde{ε}}_{(α)}) / ω^{2}$ , with positive infinitesimal constants ${\tilde{ε}}_{(α)} \to 0$ regularizing the denominators of ${\hat{G}}_{μ ν}^{α β}$ in (5.8) as $k^{2} - ω^{2} + m_{(α)}^{2} - i {\tilde{ε}}_{(α)}$ . The wave numbers thus admit an infinitesimal imaginary part and a massive dispersion relation $k_{(α)} \sim \sqrt{ω^{2} - m_{(α)}^{2}} + i / (2 δ_{(α)})$ , where $δ_{(α)} \sim \sqrt{ω^{2} - m_{(α)}^{2}} / {\tilde{ε}}_{(α)}$ is the diverging attenuation length of the field component α in the infinitesimally dissipative aether [20]. To find the lifetime associated with this attenuation length, we put ω=E−iΓ_(α)(E)/2 $ω = E - i Γ_{(α)} (E) / 2$ , and determine the decay rate Γ_(α) $Γ_{(α)}$ by solving $Im k_{(α)} (ω) = 0$ , obtaining $Γ_{(α)} \sim {\tilde{ε}}_{(α)} / E$ in linear order in the infinitesimal ${\tilde{ε}}_{(α)}$ . In this way, one can switch between complex space and time components of the plane-wave phase, using either k_(α)k₀x−ωt $k_{(α)} k_{0} x - ω t$ or $\sqrt{E^{2} - m_{(α)}^{2}} k_{0} x - (E - i {\tilde{ε}}_{(α)} / (2 E)) t$ (where we may again write ω for E). Thus the squared wave function scales with exp(−k₀x/δ_(α)) $\exp (- k_{0} x / δ_{(α)})$ or $\exp (- {\tilde{ε}}_{(α)} t / ω)$ . The lifetime $1 / Γ_{(α)} \sim ω / {\tilde{ε}}_{(α)}$ of a mode in the aether is related to its attenuation length by δ_(α)∼υ_gr(α)/Γ_(α) $δ_{(α)} \sim υ_{gr (α)} / Γ_{(α)}$ , where $υ_{gr (α)} = \sqrt{ω^{2} - m_{(α)}^{2}} / ω$ is the group velocity. At ${\tilde{ε}}_{(α)} = 0$ , the permeabilities are real and symmetric on the real axis, satisfying $ε_{(α)}^{⁎} (ω) = ε_{(α)} (- ω^{⁎})$ and $μ_{(α)}^{⁎} (ω) = μ_{(α)} (- ω^{⁎})$ in the complex plane. The antipropagator ${\hat{G}}_{μ ν}^{α β ⁎}$ is found by $δ {\hat{A}}_{μ}^{α}$ variation of L in (5.6).

To obtain the Dirac propagator, we start with the Lagrangian of a spinor multiplet ψ_j $ψ_{j}$ ,

equation5.12

L = \frac{1}{i} (\frac{1}{2} {\bar{ψ}}_{j} γ_{μ} g_{(j)}^{μ ν} ψ_{j, ν} - \frac{1}{2} {\bar{ψ}}_{j, μ} g_{(j)}^{μ ν} γ_{ν} ψ_{j} + m_{(j)} {\bar{ψ}}_{j} ψ_{j}),

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where

{\bar{ψ}}_{j} = - ψ_{j}^{†} γ_{0}

is the adjoint spinor, and

g_{(j)}^{μ ν} (ω)

the permeability tensor with components

g_{(j)}^{00} = - ε_{(j)}

g_{(j)}^{m n} = δ^{m n} / μ_{(j)}

g_{(j)}^{0 n} = 0

, see after (2.1). The Dirac equation reads

g_{(j)}^{μ ν} γ_{μ} ψ_{j, ν} + m_{(j)} ψ_{j} = 0

, where j is a multi-index labeling flavor, color, etc. We write j in parentheses if it is not meant as a summation index, cf. after (5.1). On substituting plane waves

ψ_{j} = {\hat{ψ}}_{j} e^{i (kx - ω t)}

, we find

L = - i {\bar{\hat{ψ}}}_{i} {\hat{G}}_{i j}^{(- 1)} {\hat{ψ}}_{j}

, with kernel

{\hat{G}}_{i j}^{(- 1)} = δ^{i j} (i g_{(j)}^{μ ν} k_{μ} γ_{ν} + m_{(j)})

. To invert

{\hat{G}}_{i j}^{(- 1)}

, we note γ_μγ_ν=η_μν+σ_μν

γ_{μ} γ_{ν} = η_{μ ν} + σ_{μ ν}

, where σ_μν

σ_{μ ν}

is antisymmetric, so that the product of

i g_{(j)}^{μ ν} k_{μ} γ_{ν} \pm m_{(j)}

- (h_{(j)}^{α β} k_{α} k_{β} + m_{(j)}^{2})

, with the squared permeability tensor

h_{(j)}^{α β} = g_{(j)}^{α μ} g_{(j)}^{β ν} η_{μ ν}

[10]. Hence,

equation5.13

{\hat{G}}^{i j} = - δ^{i j} \frac{i g_{(j)}^{μ ν} k_{μ} γ_{ν} - m_{(j)}}{h_{(j)}^{α β} k_{α} k_{β} + m_{(j)}^{2}} = - i δ^{i j} μ_{(j)} ω \frac{ε_{(j)} μ_{(j)} γ_{0} + γ_{n} k_{n} / ω + i μ_{(j)} m_{(j)} / ω}{k^{2} - ε_{(j)}^{2} μ_{(j)}^{2} ω^{2} + μ_{(j)}^{2} m_{(j)}^{2}},

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where we used the isotropic permeability tensor

g_{(j)}^{μ ν}

stated after (5.12). We note that

k_{n} / ω = k_{0} ε_{(j)} μ_{(j)} (1 + O (m_{(j)}^{2} / ω^{2}))

, cf. (2.1).

We drop all mass terms in (5.12) and (5.13), that is we use the massless Dirac equation, cf. after (5.12), and specify the refractive indices n_r(j)=ε_(j)μ_(j) $n_{r (j)} = ε_{(j)} μ_{(j)}$ of the multiplet ψ_j $ψ_{j}$ as, cf. (3.4),

equation5.14

n_{r (j)} = ε_{(j)} μ_{(j)} = ε (ω) μ (ω) \sqrt{1 - \frac{m_{(j)}^{2} - i {\tilde{ε}}_{(j)}}{ε^{2} (ω) ω^{2}}} .

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We also put μ_(j)=μ(ω)

μ_{(j)} = μ (ω)

and assume ε(ω)

ε (ω)

and μ(ω)

μ (ω)

to be close to one, infinitesimal

{\tilde{ε}}_{(j)}

, as well as m_(j)/ω≪1

m_{(j)} / ω ≪ 1

, so that the permeability tensor

g_{(j)}^{μ ν}

(stated after (5.12)) is close to the Minkowski metric η^μν

η^{μ ν}

. We thus find the propagator

{\hat{G}}_{0}^{i j}

of the massless Dirac equation as, cf. (5.13),

equation5.15

{\hat{G}}_{0}^{i j} = - δ^{i j} \frac{i g_{(j)}^{μ ν} k_{μ} γ_{ν}}{h_{(j)}^{α β} k_{α} k_{β}} = - δ^{i j} \frac{i μ ω (γ_{0} n_{r (j)} + γ_{n} k_{n} / ω)}{k^{2} - μ^{2} (ε^{2} ω^{2} - m_{(j)}^{2} + i {\tilde{ε}}_{(j)})} .

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In the nominator of (5.15), we can put

{\tilde{ε}}_{(j)} = 0

in n_r(j)

n_{r (j)}

. The

{\tilde{ε}}_{(j)}

are positive infinitesimal constants defining the Green function (here the propagator), i.e. the integration path around its pole; the complex wave numbers k_(j)=ωn_r(j)

k_{(j)} = ω n_{r (j)}

result in infinitesimally damped wave fields, cf. before (5.12).

In propagator ${\hat{G}}^{i j}$ of the massive Dirac equation, cf. (5.13), we put ε_(j)=ε(ω),μ_(j)=μ(ω) $ε_{(j)} = ε (ω), μ_{(j)} = μ (ω)$ , and use the integration path prescription $i {\tilde{ε}}_{(j)}$ as in (5.15). Since the neutrino mass is much smaller than the neutrino energy, cf. the end of Section 3, we can drop the iμ_(j)m_(j)/ω $i μ_{(j)} m_{(j)} / ω$ term in (5.13) and expand $n_{r (j)} = ε μ (1 + O (m_{(j)}^{2} / ω^{2}))$ in (5.14), so that the propagators (5.13) and (5.15) coincide in leading order in an m_(j)/ω $m_{(j)} / ω$ expansion. The mass-square determining the pole singularity of ${\hat{G}}^{i j}$ is recovered in propagator ${\hat{G}}_{0}^{i j}$ without invoking a mass term in the Dirac Lagrangian (5.12).

6. Dirac equation and permeability tensor in moving inertial frames: Michelson–Morley experiments with neutrino beams

We consider an inertial frame uniformly moving in the aether with velocity υ_r=υ_rυ_r,0 $υ_{r} = υ_{r} υ_{r, 0}$ . Unit vectors are denoted by a zero subscript. The spacetime coordinates in the rest frame of the aether (aether frame) are denoted by x^μ=(t,x) $x^{μ} = (t, x)$ and in the inertial frame by y^μ=(τ,y) $y^{μ} = (τ, y)$ . The proper orthochronous Lorentz boost $x^{μ} = Λ_{ν}^{μ} y^{ν}$ relating the frames is

equation6.1

Λ_{0}^{0} = γ_{r}, Λ_{k}^{0} = \sqrt{γ_{r}^{2} - 1} υ_{r, 0}^{k}, Λ_{0}^{i} = \sqrt{γ_{r}^{2} - 1} υ_{r, 0}^{i},

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Λ_{k}^{i} = δ_{i k} + (γ_{r} - 1) υ_{r, 0}^{i} υ_{r, 0}^{k}, γ_{r} = {(1 - υ_{r}^{2})}^{- 1 / 2} .

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The inverse transformation

y^{μ} = Λ_{ν}^{(- 1) μ} x^{ν}

is found by changing the sign of the relative velocity υ_r,0

υ_{r, 0}

. Contravariant quantities transform from the aether frame to a moving inertial frame like

g_{inertial}^{μ ν} = g_{aether}^{α β} Λ_{α}^{(- 1) μ} Λ_{β}^{(- 1) ν}

, covariant ones like

g_{μ ν}^{in} = g_{α β}^{ae} Λ_{μ}^{α} Λ_{ν}^{β}

, and mixed quantities according to this pattern. Indices are raised and lowered with the invariant Minkowski metric η_μν=diag(−1,1,1,1)

η_{μ ν} = diag (- 1, 1, 1, 1)

. Energy and momentum variables are denoted by k_μ=(−ω,k)

k_{μ} = (- ω, k)

in the aether frame and by p_μ=(−E,p)

p_{μ} = (- E, p)

in the inertial frame, and

p_{μ} = k_{α} Λ_{μ}^{α}

In the aether frame, the Dirac equation (1.1) reads

equation6.2

(γ_{μ} g_{ae}^{μ ν} (ω) \frac{\partial}{\partial x^{ν}} + m) ψ_{ae} (ω; x) = 0,

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coupled to the isotropic permeability tensor

g_{ae}^{μ ν} (ω)

in (1.2). We consider plane waves

ψ_{ae} (ω; x) = {\tilde{ψ}}_{ae} e^{i k_{μ} x^{μ}}

, where

{\tilde{ψ}}_{ae}

is a constant 4-spinor. The wave vector k_μ=(−ω,k)

k_{μ} = (- ω, k)

, k=k(ω)k₀

k = k (ω) k_{0}

, satisfies the dispersion relation (2.1), k²=μ²(ε²ω²−m²)

k^{2} = μ^{2} (ε^{2} ω^{2} - m^{2})

In the moving inertial frame related by the Lorentz boost (6.1) to the aether frame, the permeability tensor is anisotropic, $g_{in}^{μ ν} (ω) = g_{ae}^{α β} (ω) Λ_{α}^{(- 1) μ} Λ_{β}^{(- 1) ν}$ . The tensors $g_{ae, in}^{μ ν} (ω)$ are independent of space and time coordinates; they only depend on the frequency ω of the wave modes in the aether frame, which is the universal frame of reference. We denote the boosts (6.1) by x=Λ(y) $x = Λ (y)$ and y=Λ⁻¹(x) $y = Λ^{- 1} (x)$ . To find the Dirac equation in the inertial frame, we write Eq. (6.2) as

equation6.3

(γ_{μ} Λ_{α}^{μ} g_{in}^{α β} (ω) \frac{\partial}{\partial y^{β}} + m) ψ_{ae} (ω; Λ (y)) = 0,

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and consider a spinor field ψ_in(ω;y)

ψ_{in} (ω; y)

in the inertial frame, related to a plane-wave solution ψ_ae(ω;x)

ψ_{ae} (ω; x)

of Eq. (6.2) by a similarity S_Λ

S_{Λ}

, so that ψ_in(ω;y)=S_Λψ_ae(ω;Λ(y))

ψ_{in} (ω; y) = S_{Λ} ψ_{ae} (ω; Λ (y))

and

ψ_{ae} (ω; x) = S_{Λ}^{- 1} ψ_{in} (ω; y)

. We may thus write Eq. (6.3) as

equation6.4

(S_{Λ} γ_{μ} S_{Λ}^{- 1} Λ_{α}^{μ} g_{in}^{α β} (ω) \frac{\partial}{\partial y^{β}} + m) ψ_{in} (ω; y) = 0 .

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By choosing the matrix S_Λ

S_{Λ}

as a solution of

S_{Λ} γ_{μ} S_{Λ}^{- 1} = γ_{α} Λ_{μ}^{(- 1) α}

equation6.5

S_{Λ} = \frac{1}{\sqrt{2} \sqrt{γ_{r} - 1}} (\sqrt{γ_{r}^{2} - 1} + (1 - γ_{r}) γ_{0} γ_{i} υ_{r, 0}^{i}),

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where γ_r

γ_{r}

denotes the Lorentz factor, cf. (6.1), we find the Dirac equation in the inertial frame,

equation6.6

(γ_{α} g_{in}^{α β} (ω) \frac{\partial}{\partial y^{β}} + m) ψ_{in} (ω; y) = 0 .

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The inverse similarity

S_{Λ}^{- 1}

is obtained by changing the sign of the velocity unit vector in (6.5), so that

S_{Λ}^{- 1} = S_{Λ^{- 1}}

. The Dirac matrices in (6.5) satisfy γ_μγ_ν+γ_νγ_μ=2η_μν

γ_{μ} γ_{ν} + γ_{ν} γ_{μ} = 2 η_{μ ν}

, where γ₀

γ_{0}

is anti-Hermitian and the γ_n

γ_{n}

and S_Λ

S_{Λ}

are Hermitian, and

γ_{0} S_{Λ} = S_{Λ}^{- 1} γ_{0}

. A plane-wave solution

ψ_{ae} (ω; x) = {\tilde{ψ}}_{ae} e^{i k_{μ} x^{μ}}

of (6.2) in the aether frame is transformed into

ψ_{in} (ω; y) = {\tilde{ψ}}_{in} e^{i p_{μ} y^{μ}}

solving (6.6), where

{\tilde{ψ}}_{in} = S_{Λ} {\tilde{ψ}}_{ae}

. The dispersion relations in the aether frame and the inertial frame can be written as

h_{ae}^{μ ν} k_{μ} k_{ν} + m^{2} = 0

and

h_{in}^{μ ν} p_{μ} p_{ν} + m^{2} = 0

, respectively, where

h_{ae, in}^{μ ν} (ω) = g_{ae, in}^{μ α} η_{α β} g_{ae, in}^{β ν}

, cf. (5.13). The Dirac equation in (6.2) and (6.6) has a manifestly covariant appearance as well, but is not covariant at all, since the anisotropic permeability tensor

g_{in}^{μ ν} (ω)

of the inertial frame depends on the energy ω of the wave field in the aether frame, which is the universal frame of reference;

g_{in}^{μ ν}

is obtained by applying a Lorentz boost to the isotropic reference tensor

g_{ae}^{μ ν}

of the aether frame.

As for Michelson–Morley experiments, we consider neutrinos propagating with constant (group) velocity υ_gr=υ_grυ_gr,0 $υ_{gr} = υ_{gr} υ_{g r, 0}$ in the aether frame where the permeability tensor is isotropic. An inertial laboratory frame moves with constant relative speed υ_r=υ_rυ_r,0 $υ_{r} = υ_{r} υ_{r, 0}$ in the aether. The frames are related by the Lorentz boost (6.1). In the context of neutrino velocity experiments [6], [7], [8] and [9], the inertial frame is the neutrino baseline frame (rest frame of neutrino source and detector), slowly moving with a relative velocity of order υ_r=O(10⁻³) $υ_{r} = O (10^{- 3})$ in the aether, cf. the discussion after (6.10). In the moving inertial frame, the neutrino velocity is denoted by υ_in=υ_inυ_in,0 $υ_{in} = υ_{in} υ_{i n, 0}$ and related to the group velocity in the aether frame by the relativistic addition law (linearized in the relative speed υ_r $υ_{r}$ ),

equation6.7

υ_{in} \sim υ_{gr} (1 + (υ_{gr}^{2} - 1) \frac{υ_{r}}{υ_{gr}} \cos θ_{gr}),

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\cos θ_{in} \sim \cos θ_{gr} - \frac{υ_{r}}{υ_{gr}} \sin^{2} θ_{gr},

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and inversely,

equation6.8

υ_{gr} \sim υ_{in} (1 - (υ_{in}^{2} - 1) \frac{υ_{r}}{υ_{in}} \cos θ_{in}),

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cosθgr∼cosθin+υrυinsin2θin, $\cos θ_{gr} \sim \cos θ_{in} + \frac{υ_{r}}{υ_{in}} \sin^{2} θ_{in},$

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where

cosθgr=υgr,0υr,0 $\cos θ_{gr} = υ_{g r, 0} υ_{r, 0}$ and

cosθin=υin,0υr,0 $\cos θ_{in} = υ_{i n, 0} υ_{r, 0}$ , which applies in slowly moving inertial frames, υ_r≪1υr≪1 $υ_{r} ≪ 1$ . The velocities υ_grυgr $υ_{gr}$ and υ_inυin $υ_{in}$ are either both sub- or superluminal, which can be seen from the exact velocity addition law [10]. The transformation (6.7) and its inverse (6.8) are complemented by the Doppler shift connecting the neutrino energy E measured in the inertial frame to the energy ω of the wave mode in the aether frame according to

pμ=kαΛαμ $p_{μ} = k_{α} Λ_{μ}^{α}$ , cf. after (6.1). The Doppler shift (linearized in υ_rυr $υ_{r}$ ) and its inverse read

equation6.9

E∼ω(1−n_r(ω)υ_rcosθ_gr),E∼ω(1−nr(ω)υrcosθgr), $E \sim ω (1 - n_{r} (ω) υ_{r} \cos θ_{gr}),$

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ω∼E(1+n_r(E)υ_rcosθ_in),ω∼E(1+nr(E)υrcosθin), $ω \sim E (1 + n_{r} (E) υ_{r} \cos θ_{in}),$

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where n_r(ω)nr(ω) $n_{r} (ω)$ is the refractive index

equation6.10

nr(ω)=k(ω)ω=με1−m2ε2ω2−−−−−−√,1υgr=(ωnr(ω))′, $n_{r} (ω) = \frac{k (ω)}{ω} = μ ε \sqrt{1 - \frac{m^{2}}{ε^{2} ω^{2}}}, \frac{1}{υ_{gr}} = {(ω n_{r} (ω))}^{'},$

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defined by the dispersion relation in the aether frame, cf. after (6.2). n_r(E)nr(E) $n_{r} (E)$ in (6.9) means n_r(ω)nr(ω) $n_{r} (ω)$ taken at ω=Eω=E $ω = E$ . (Terms quadratic in υ_rυr $υ_{r}$ are neglected in (6.9).) The Doppler shift from the aether frame to a moving inertial frame depends not only on the relative velocity but also on the refractive index, which is a function of the neutrino frequency ω in the aether frame. Due to the dispersive permeabilities, the neutrino group velocity υ_gr(ω)υgr(ω) $υ_{gr} (ω)$ is energy dependent even for massless neutrinos.

The group velocity υ_grυgr $υ_{gr}$ in the aether frame is obtained via transformation (6.8), once we have measured the neutrino velocity υ_inυin $υ_{in}$ in the inertial laboratory frame and the angle cosθin $\cos θ_{in}$ between inertial velocity and relative speed. Assuming adiabatic frequency variation of the refractive index, ωn′r(ω)/nr≪1 $ω n_{r}^{'} (ω) / n_{r} ≪ 1$ , we can identify n_r∼1/υ_grnr∼1/υgr $n_{r} \sim 1 / υ_{gr}$ , cf. (6.10). The neutrino frequency ω in the aether frame is then found by measuring the neutrino energy E in the inertial frame, ω/E∼1+(υr/υgr)cosθin $ω / E \sim 1 + (υ_{r} / υ_{gr}) \cos θ_{in}$ , cf. (6.9). Adiabatic variation of n_r(ω)nr(ω) $n_{r} (ω)$ requires adiabatic permeabilities, ωε^′(ω)/ε≪1ωε′(ω)/ε≪1 $ω ε^{'} (ω) / ε ≪ 1$ and ωμ^′(ω)/μ≪1ωμ′(ω)/μ≪1 $ω μ^{'} (ω) / μ ≪ 1$ , as well as a small neutrino mass, m/ω≪1m/ω≪1 $m / ω ≪ 1$ , cf. (6.10).

The neutrino velocity υ_inυin $υ_{in}$ in the laboratory frame depends on the angle cosθin $\cos θ_{in}$ between the neutrino velocity and the relative velocity of the inertial frame in the aether, cf. (6.7). The minimal (maximal) neutrino velocity υ_inυin $υ_{in}$ is attained at θ_in=0θin=0 $θ_{in} = 0$ or θ_in=πθin=π $θ_{in} = π$ , when the velocity vectors are aligned or opposite, depending on whether the refractive index n_r∼1/υ_grnr∼1/υgr $n_{r} \sim 1 / υ_{gr}$ is larger or smaller than one (sub- or superluminal group velocity in the aether frame). The maximal variation of the inertial neutrino speed in the baseline frame is thus δυin∼2υr(1/n2r−1) $δ υ_{in} \sim 2 υ_{r} (1 / n_{r}^{2} - 1)$ , cf. (6.7). If the neutrino beam can arbitrarily be rotated and the neutrino velocity υ_inυin $υ_{in}$ is measured as a function of the angles parametrizing the rotation, one can in this way determine the relative speed υ_rυr $υ_{r}$ of the inertial frame (here the local baseline rest frame of source and detector) in the aether, as attempted in the optical Michelson–Morley experiment. At θ_in=π/2θin=π/2 $θ_{in} = π / 2$ , when the velocities υ_rυr $υ_{r}$ and υ_inυin $υ_{in}$ are orthogonal, the measured inertial neutrino speed υ_inυin $υ_{in}$ coincides with the neutrino group velocity υ_gr∼1/n_rυgr∼1/nr $υ_{gr} \sim 1 / n_{r}$ in the aether frame (that is, the linear order in the relative speed υ_rυr $υ_{r}$ vanishes, cf. (6.8)). The OPERA Collaboration derived the bound υ_in−1=(2.7±6.5)×10⁻⁶ $υ_{in} - 1 = (2.7 \pm 6.5) \times 10^{- 6}$ on the neutrino velocity [6]. To illustrate the angular dependence of this velocity, we use υ_in−1≈2.7×10⁻⁶ $υ_{in} - 1 \approx 2.7 \times 10^{- 6}$ and the relative speed υ_r≈1.2×10⁻³ $υ_{r} \approx 1.2 \times 10^{- 3}$ of the Solar system barycenter in the cosmic microwave background. (Earthʼs relative velocities in the Solar system are by at least one order smaller.) We note υ_gr−1=(υ_in−1)(1+O(υ_r)) $υ_{gr} - 1 = (υ_{in} - 1) (1 + O (υ_{r}))$ , cf. (6.8), and find a maximal angular variation of δυ_in≈4υ_r(υ_in−1)≈1.3×10⁻⁸ $δ υ_{in} \approx 4 υ_{r} (υ_{in} - 1) \approx 1.3 \times 10^{- 8}$ for the neutrino velocity in the inertial frame, which is still by three orders smaller than the accuracy of current υ_in $υ_{in}$ measurements [7], [8], [9] and [26]. The isotropic aether frame is identified as the rest frame of the microwave background defined by vanishing temperature dipole anisotropy [12] and [13].

7. Conclusion

We demonstrated that effective mass-squares can be generated by permeability tensors in spinor as well as gauge fields, for fermions and bosons alike, familiar from the electrodynamics of dispersive dielectric media. This analogy is evident in each step, from the Lagrangians to the propagators, cf. Sections 4 and 5. Massless gauge fields coupled to the aether by dispersive permeability tensors admit massive dispersion relations preserving the gauge invariance. Mass generation by a permeability tensor is particularly attractive in the case of neutrino mixing, when small masses in the sub-eV range are needed [27] and [28], since such masses are generated by permeability tensors close to the Minkowski metric, cf. Sections 2 and 5. Interference factors in the flavor transition probabilities are due to phase differences of the mass-eigenstate plane waves, which result in oscillations if the neutrino path length is sufficiently large. All neutrino flavors propagate at the same group velocity, υ_gr≈1/(εμ) $υ_{gr} \approx 1 / (ε μ)$ , as the refractive indices of high-energy mass eigenstates only differ by negligible terms of order $m_{(j)}^{2} / ω^{2}$ , cf. Section 3. Sub- and superluminal neutrino velocities close to the speed of light can be described on equal footing by permeability tensors; the permeabilities defining the dispersion relations are slightly larger than one in the case of subluminal neutrino speeds and smaller than one for superluminal neutrinos. The neutrino velocity measured in the baseline frame (rest frame of neutrino source and detector) depends on the angle between the neutrino beam (baseline) and its relative velocity in the aether. This angular dependence can be used to locally determine the velocity of source and detector in the aether, cf. Section 6, without invoking the temperature anisotropy of the cosmic microwave background in the neutrino baseline frame.

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  Bibliographic information

Mass generation in the aether: Neutrino oscillations and massive gauge fields

Abstract

Highlights

Keywords

1. Introduction

2. Neutrino mixing in a dispersive spacetime

3. Mass eigenstates generated by permeability tensors

4. Gauge and Proca fields in the aether: Massive dispersion relations preserving gauge invariance

5. Effective mass-squares in propagators of dispersive gauge and Dirac fields

6. Dirac equation and permeability tensor in moving inertial frames: Michelson–Morley experiments with neutrino beams

7. Conclusion

References

Bibliographic information