Obtaining consistent Lorentz gauging for a gravitationally coupled fermion

TL;DR

This paper develops a formalism for gravitationally coupled fermions that maintains the consistency of local gauge transformations, avoiding the usual violations of the equivalence principle in the vierbein formalism.

Contribution

It introduces a new approach where the fermion field carries a world index instead of Lorentz indices, ensuring gauge transformation commutativity in gravitational coupling.

Findings

Achieves gauge invariance consistency for fermions in gravity

Maintains Einstein-Hilbert action for gravity

Simplifies fermion coupling by removing Lorentz indices

Abstract

For internal gauge forces, the result of locally gauging, i.e., of performing the substitution $\partial \rightarrow D$, is physically the same whether performed on the action or on the corresponding Euler-Lagrange equations of motion. Rather unsettling, though, such commutativity fails for the standard way of coupling a Dirac fermion to the gravitational field in the setting of a local Lorentz gauge theory of general relativity in the vierbein formalism, the equivalence principle thus seemingly being here violated. This paper will present a formalism in which commutativity holds for the gravitational force as well, the action for the gravitational field itself being still the Einstein-Hilbert one. Notably, in this formalism, the spinor field will carry a world/coordinate index, rather than a Lorentz spinor index as it does standardly. More generally, no Lorentz indices will figure, neither vector indices nor spinor indices, which from a parsimonious point of view seems quite satisfactory.

Equations

$$S_{free}$$

$${\mathcal{L}}_{free}$$

$$S$$

$$\mathcal{L}$$

$${\partial }_{\mu }\psi \to D_{\mu }\psi \equiv \left({\partial }_{\mu }+\frac{1}{2}{\omega }_{\mu ab}{S}^{ab}\right)\psi ,$$

$$S_{grav}$$

$${\mathcal{L}}_{grav}$$

$$0$$

$$1$$

$$0$$

$$-{\delta }_{ij}$$

$$g_{\mu \nu }$$

$$\delta n^{\mu }$$

$$\delta {\overline{n}}^{\mu }$$

$$-\left[{\mathbf{V}}^{\mu \nu },{\mathbf{V}}^{\rho \sigma }\right]=g^{\mu \rho }{\mathbf{V}}^{\nu \sigma }-g^{\mu \sigma }{\mathbf{V}}^{\nu \rho }-g^{\nu \rho }{\mathbf{V}}^{\mu \sigma }+g^{\nu \sigma }{\mathbf{V}}^{\mu \rho },$$

$$\delta n^{\mu }$$

$$\delta {\overline{n}}^{\mu }$$

$$-d{\theta }_{\alpha \beta }=\left({\overline{n}}_{\alpha }\times {\overline{n}}_{\beta }\right)\cdot \overline{d\theta }+\left(n_{\alpha }{\overline{n}}_{\beta }-n_{\beta }{\overline{n}}_{\alpha }\right)\cdot \overline{d\xi }.$$

$$0$$

$$\overline{0}$$

$${\omega }^{\mu }{}_{\nu \rho }$$

$$\equiv$$

$$\delta {\omega }^{\mu }{}_{\nu \rho }=\left(d{\theta }^{\mu }{}_{\sigma }\right){\omega }^{\sigma }{}_{\nu \rho }-\left(d{\theta }^{\sigma }{}_{\nu }\right){\omega }^{\mu }{}_{\sigma \rho }-{\nabla }_{\rho }d{\theta }^{\mu }{}_{\nu }.$$

$$\delta {\omega }_{\rho }{}^a{}_b=\left(d{\theta }^a{}_c\right){\omega }_{\rho }{}^c{}_b-\left(d{\theta }^c{}_b\right){\omega }_{\rho }{}^a{}_c-{\partial }_{\rho }d{\theta }^a{}_b,$$

$$\delta \left(D_{\rho }{n}^{\mu }\right)$$

$$\delta \left(D_{\rho }{\overline{n}}^{\mu }\right)$$

$$0$$

$$\overline{0}$$

$$0$$

$$\overline{0}$$

$$R^{\mu }{}_{\nu \rho \sigma }$$

$${\Omega }^{\mu }{}_{\nu \rho \sigma }$$

$$0$$

$${\left({\mathbf{V}}^{\hat{\mathrm{T}}}\right)}_{\mu }$$

$${\left({\mathbf{A}}^{\hat{\mathrm{T}}}\right)}^{\mu }{}_{\nu }$$

$${\mathbf{V}}^{\widehat{†}}$$

Full text

1 Introduction

Consider in global Minkowski spacetime, with metric $\eta_{ab}$ in Euclidian coordinates $x^{a}$, the following free Dirac action (written in explicitly hermitian form):

[TABLE]

Here, the factor $\sqrt{-\eta}=1$, although trivial, has been included for completeness. The corresponding Euler-Lagrange equations of motion are given by $E_{free}\equiv\left(\mathrm{i}\gamma^{a}\partial_{a}-m\right)\psi=0$. Let $\psi$ have an electric charge $q$, say. Then in the presence of an external electromagnetic field $A_{a}$, the Lagrangian is augmented to

[TABLE]

where $D_{a}=\partial_{a}+\mathrm{i}qA_{\mu}$. The corresponding Euler-Lagrange equations of motion are now given by $E\equiv\left(\mathrm{i}\gamma^{a}D_{a}-m\right)\psi=0$. Reassuringly, $E$ results whether the substitution $\partial_{a}\rightarrow D_{a}$ is performed on $S_{free}$ or on $E_{free}$; the substitution procedure $\partial_{a}\rightarrow D_{a}$ may thus be said to commute with the Euler-Lagrange variational procedure. This commutativity property holds not only for an electromagnetically coupled fermion; it holds as well for a weakly coupled fermionic doublet, and for a strongly coupled fermionic triplet, the reason being that the generators for the weak and strong forces, respectively, commute with $1_{2}\otimes\gamma^{a}$ and $1_{3}\otimes\gamma^{a}$, where $1_{n}$ means the $n\times n$ unit matrix. Generally, it holds for any internal gauge force with generators commuting with $1_{n}\otimes\gamma^{a}$ (for appropiate values of $n$).

All this would be pretty uninteresting, though, was it not for the following fact: such commutativity fails for a standardly gravitationally coupled fermion. Proof: To switch on gravitational and/or inertial forces, in the realm of general relativity recasted as a local Lorentz gauge theory, the standard procedure, compare [1, Sec. 31.A] and [2, Sec. 12.1], is 1.) to introduce a vierbein, $e^{\mu}{}_{a}$, and an associated minimal (i.e., torsionless) spin connection, $\omega_{\mu}{}^{a}{}_{b}\equiv e^{a}{}_{\rho}\nabla_{\mu}e^{\rho}{}_{b}$, and 2.) to perform in conjunction the substitutions $\eta_{ab}\rightarrow g_{\mu\nu}=\eta_{ab}e^{a}{}_{\mu}e^{b}{}_{\nu}$ and $\gamma^{a}\rightarrow\gamma^{\mu}=e^{\mu}{}_{a}\gamma^{a}$ and

[TABLE]

where $S^{ab}\equiv\frac{1}{4}\left[\gamma^{a},\gamma^{b}\right]$ are the generators of the spinor representation of the Lorentz group, being here defined in the ’mathematicians way’ without an explicit $\mathrm{i}$. Applied to the free action $S_{free}$ previously given, the result is

[TABLE]

for a Dirac fermion in an external gravitational field. The corresponding Euler-Lagrange equations of motion are given by

[TABLE]

using 1.) the identity $\partial_{\mu}\left(e^{\mu}{}_{a}\sqrt{-g}\right)=\nabla_{\mu}e^{\mu}{}_{a}\equiv\partial_{\mu}e^{\mu}{}_{a}+\Gamma^{\mu}{}_{\rho\mu}e^{\rho}{}_{a}$, where $\Gamma^{\mu}{}_{\nu\rho}$ is the Levi-Civita connection, and 2.) the minimality of the spin connection. But if the substitution is instead applied to the Euler-Lagrange equations of motion $E_{free}$ previously given, the result is

[TABLE]

End of proof. This nonequality of $E_{grav}$ and $\tilde{E}_{grav}$ seems to the author rather unsettling as it seems to imply that the equivalence principle is violated (for a Dirac fermion): Whereas $\tilde{E}_{grav}$ seems to be the correct way of implementing the equivalence principle, $E_{grav}\neq\tilde{E}_{grav}$ derived from an action by Euler-Lagrange variation must necessarily take precedence over it, thus resulting in an inconsistency.

The main purpose of this paper is to present a formalism for the coupling of a fermion to the gravitational field in which no such ambiguity arises, i.e., in which the substitution procedure $\partial\rightarrow D$, now with a different $D$ of course, commute with the Euler-Lagrange variational procedure. The formalism will contain only world indices, with neither Lorentz vector indices nor Lorentz spinor indices figuring; contrary to what appears to be standard wisdom, it will prove possible to have the spinor field carry a world index rather than a Lorentz spinor index.

2 Preliminaries, I: Geometry

Let $\left(M,g,\Gamma\right)$ be a Riemannian manifold $M$ equipped with a metric $g$ of signature $\left(1,3\right)$ and corresponding Levi-Civita connection $\Gamma$. Introduce on this manifold one timelike- and three spacelike vector fields, $n^{\mu}$ and $n_{i}^{\mu}$, respectively, subject to the following conditions:

[TABLE]

[TABLE]

introducing the three-vector of four-vectors $\overline{n}_{\mu}$ by $\left(\overline{n}_{\mu}\right)_{i}\equiv n_{i\mu}$, the dot product being performed over the Latin indices. Here, and below, a bar will denote a three-vector quantity. As for the standard vierbein, this expression for the metric in terms of four vector fields introduces excess local degrees of freedom: the metric is invariant under the following local Lorentz transformations:

[TABLE]

[TABLE]

i.e., they are the generators of the vector representation of the Lorentz group.

A remark: Strictly speaking, the above transformation is not a local Lorentz transformation, as it operates on world indices, rather than on Lorentz vector indices. But it may, nonetheless, by a mild abuse of terminology (which will be adhered to in the rest of the paper), be called so for the following reason: A genuine (infinitesimal) local Lorentz transformation, not acting on any world indices, is given by

[TABLE]

[TABLE]

The overall minus sign in this relationship is due to the fact that the Lorentz transformations of Eqs. (4a)-(4b) act on contravariant world indices, whereas the Lorentz transformations of Eqs. (6a)-(6b) act on covariant Lorentz indices (the $a$ of $e^{\mu}{}_{a}$, remembering the previously mentioned possible identifications $e^{\mu}{}_{0}=n^{\mu}$ and $e^{\mu}{}_{i}=n_{i}^{\mu}$). End of remark.

As for the standard vierbein formulation of general relativity, compare again [1, Sec. 31.A] and [2, Sec. 12.1], these excess Lorentz degrees of freedom should be killed in order to avoid augmenting the standard content of general relativity. As standardly, this is done by requirering that the local Lorentz frame field consisting of $n^{\mu}$ and $\overline{n}^{\mu}$ in conjunction be covariantly constant:

[TABLE]

for some connection $\omega^{\mu}{}_{\nu\rho}$ to be introduced. The unique solution to these conditions is

[TABLE]

[TABLE]

[A remark: The presence of the Levi-Civita covariant derivative in Eq. (10) makes $\delta\omega^{\mu}{}_{\nu\rho}$ a type $\left(1,2\right)$ world tensor. By a continuing mild abuse of terminology, compare previous remark, it is also seen to be a type $\left(1,1\right)$ Lorentz tensor in the indices $\mu\nu$, just as the expression

[TABLE]

for the standard spin connection is a type $\left(1,1\right)$ Lorentz tensor in the indices $ab$. End of remark.] It is readily established that

[TABLE]

as is appropiate for a proper covariant derivative. These relations say that $D_{\rho}n^{\mu}$ and $D_{\rho}\overline{n}^{\mu}$ are each type $\left(1,0\right)$ Lorentz tensors (in the index $\mu$). Therefore

[TABLE]

from which it follows that

[TABLE]

where

[TABLE]

introducing $\Omega^{\mu}{}_{\nu\rho\sigma}$. Here, $R^{\mu}{}_{\nu\rho\sigma}$ is of course the standard Riemann curvature tensor in terms of the Levi-Civita connection. [A remark: Although $R^{\mu}{}_{\nu\rho\sigma}$ and $\Omega^{\mu}{}_{\nu\rho\sigma}$ have closely analogous structure, the following difference should be noted: Whereas the Levi-Civita symbols $\Gamma^{\mu}{}_{\nu\rho}$ transform only as a tensor in the index $\mu$, the spin connection $\omega^{\mu}{}_{\nu\sigma}$ transforms as a tensor in all its indices. This explains the appearance of Levi-Civita covariant derivatives in the definition of $\Omega^{\mu}{}_{\nu\rho\sigma}$, as opposed to only the partial derivatives in the expression for $R^{\mu}{}_{\nu\rho\sigma}$. End of remark.] But then

[TABLE]

using $n^{\tau}n_{\nu}-\overline{n}^{\tau}\cdot\overline{n}_{\nu}=\delta_{\nu}^{\tau}$, from which it follows that $\Omega^{\mu}{}_{\nu\rho\sigma}=-R^{\mu}{}_{\nu\rho\sigma}$. As the Riemann tensor is locally Lorentz invariant, because the metric is so, this immediately implies that $\Omega^{\mu}{}_{\nu\rho\sigma}$ is as well. The Einstein-Hilbert action is obviously proportional to $g^{\mu\rho}g^{\nu\sigma}\Omega_{\mu\nu\rho\sigma}$.

3 Preliminaries, II: Algebra

It will prove useful to define a transposition operator $\mathrm{\hat{T}}$, say, by

[TABLE]

for any four-column vector $\mathbf{V}$, and any $4\times 4$ matrix $\mathbf{A}$. Its action would become that of the standard transposition operator $\mathrm{T}$ if $g_{\mu\nu}=\mathrm{diag}\left(1,1,1,1\right)$. Note that $\mathbf{V}^{\mathrm{\hat{T}}}$ is, as it should be, a row vector, carrying a lower/covariant index. [A remark: For any matrix, the row index will always be an upper/contravariant index, and the column index will always be a lower/covariant index, with matrix multiplication thus being given by $\left(\mathbf{AB}\right)^{\rho}{}_{\sigma}=\left(\mathbf{A}\right)^{\rho}{}_{\tau}\left(\mathbf{B}\right)^{\tau}{}_{\sigma}=A^{\rho}{}_{\tau}B^{\tau}{}_{\sigma}$, as usual, for any two matrices $\mathbf{A},\mathbf{B}$. End of remark.] It is readily proved that it shares with $\mathrm{T}$ the properties $\left(\mathbf{AB}\right)^{\mathrm{\hat{T}}}=\mathbf{B}^{\mathrm{\hat{T}}}\mathbf{A}^{\mathrm{\hat{T}}}$ and $\left(\mathbf{AV}\right)^{\mathrm{\hat{T}}}=\mathbf{V}^{\mathrm{\hat{T}}}\mathbf{A}^{\mathrm{\hat{T}}}$, for any $4\times 4$ matrices $\mathbf{A},\mathbf{B}$, and any four-column vector $\mathbf{V}$. Naturally associated with $\mathrm{\hat{T}}$ is $\hat{\dagger}$ defined by

[TABLE]

A matrix $\mathbf{A}$ for which $\mathbf{A}^{\mathrm{\hat{T}}}=\pm\mathbf{A}$ will be called hat-(anti)symmetric, and a matrix $\mathbf{A}$ for which $\mathbf{A}^{\hat{\dagger}}=\pm\mathbf{A}$ will be called hat-(anti)hermitian.

3.1 Concerning Klein-Gordon compatibility

By the notion ’Klein-Gordon compatibility’ is generally meant the requirement that all solutions to some given Euler-Lagrange equations of motion are on mass-shell. Consider the following tensorial quantities:

[TABLE]

where $\varepsilon_{\mu\nu\rho\sigma}\equiv\sqrt{-g}\left[\mu\nu\rho\sigma\right]$ is the Levi-Civita tensor in the notation of [4, Eq. (8.10a)]. Note that $N_{i\rho\sigma}$ constitutes three rank two (world) tensors, one for each value of $i$. Define the $4\times 4$ matrices $\mathbf{M}^{\mu}$ and $\mathbf{N}_{i}$ by

[TABLE]

They satisfy the following algebra (note that there is no complex conjugation of $\mathbf{M}^{\mu}$ in the second relation):

[TABLE]

where $\mathbf{1}$ and $\mathbf{0}$ are respectively the $4\times 4$ identity matrix and the $4\times 4$ zero matrix. Furthermore, the matrices $\mathbf{N}_{i}$ satisfy the following algebra:

[TABLE]

The sign in front of the Levi-Civita symbol depends on whether the three spacelike $n_{i}^{\mu}$ form a right-handed basis (plus sign) or left-handed basis (minus sign) when considered as three-vectors in the $3D$ subspace they span. Note that for the right-handed case, $\mathbf{N}_{i}$ thus obey the same algebra as do the Pauli matrices. Eqs. (19)-(21) are relevant for the proof of Klein-Gordon compatibility of some Euler-Lagrange equations of motion to be derived in Sec. 4 below. In particular, Eq. (19) will in the present formalism play a role analogous to the Dirac algebra (of the gamma matrices) in the standard Dirac algebra. The matrices $\mathbf{M}^{\mu}$ and $\mathbf{N}_{i}$ are respectively hat-hermitian and hat-antisymmetric:

[TABLE]

These two relations are relevant for the proof of hermiticity (complex self-conjugacy) of the Lagrangian to be studied in Sec. 4 below.

3.2 Concerning Lorentz invariance

Introduce the $4\times 4$ matrices $\mathbf{S}^{\mu\nu}=-\mathbf{S}^{\nu\mu}$ by

[TABLE]

where $\mathbf{M}^{\mu}$ is given by Eq. (17). They satisfy the following relations (note that there is no complex conjugation of $\mathbf{S}^{\mu\nu}$ in the second relation):

[TABLE]

These relations are readily proved using respectively Eq. (19) and Eq. (20). The proof of Eq. (25), in particular, is structurally analogous to the proof, using the Dirac algebra of gamma matrices, of the identity $\left[\gamma^{c},S^{ab}\right]=\left(V^{ab}\right)^{c}{}_{d}\gamma^{d}$ in the standard Dirac formalism, the only difference being the appearence of complex conjugations here and there. These matrices $\mathbf{S}^{\mu\nu}$ constitute the spinor representation of the Lorentz algebra in the sense that they satisfy

[TABLE]

i.e., they are the generators of the spinor representation of the Lorentz group. This algebra is readily proved using Eq. (25), the proof being structurally analogous to the proof, using $\left[\gamma^{c},S^{ab}\right]=\left(V^{ab}\right)^{c}{}_{d}\gamma^{d}$, of the fact that $S^{ab}=\frac{1}{4}\left[\gamma^{a},\gamma^{b}\right]$ in the standard Dirac formalism constitute the spinor representation of the Lorentz algebra, the only difference being, as before, the appearence of complex conjugations here and there. These generators are related to the previously introduced vector representation $\mathbf{V}^{\mu\nu}$, compare Sec. 2, as

[TABLE]

from which it follows that $\mathbf{S}^{\mu\nu}$ is self-dual, $\mathbf{S}^{\mu\nu}=\frac{\mathrm{i}}{2}\varepsilon^{\mu\nu\rho\sigma}\mathbf{S}_{\rho\sigma}$, and that

[TABLE]

using $\mathbf{V}^{\mu\nu}+\mathbf{V}^{\mu\nu\mathrm{\hat{T}}}=\mathbf{0}$ and the identity $g^{\rho\alpha}g_{\sigma\beta}\varepsilon^{\mu\nu\beta}{}_{\alpha}=-\varepsilon^{\mu\nu\rho}{}_{\sigma}$. Using Eq. (30), to switch $\mathbf{S}^{\mu\nu}$ into $\mathbf{S}^{\mu\nu\ast}$ (plus some $\mathbf{V}^{\mu\nu}$), Eqs. (25)-(26) may be rewritten as

[TABLE]

which using Eq. (29) and the definition of $\hat{\dagger}$, Eq. (14), may also be written as

[TABLE]

As the metric, and thus as well the Levi-Civita tensor, is invariant under local Lorentz transformations of $n^{\mu}$ and $\overline{n}^{\mu}$, Eqs. (4a)-(4b), these transformations of $n^{\mu}$ and $\overline{n}^{\mu}$ induce via Eqs. (15)-(16) the following relations:

[TABLE]

i.e., $M_{\mu\rho\sigma}$ and $N_{i\rho\sigma}$ transform as type $\left(0,3\right)$ and $\left(0,2\right)$ Lorentz tensors, respectively; or, equivalently, by raising various indices appropiately:

[TABLE]

i.e., $M^{\mu\rho}{}_{\sigma}$ and $\delta N_{i}{}^{\rho}{}_{\sigma}$ transform as type $\left(2,1\right)$ and $\left(1,1\right)$ Lorentz tensors, respectively. Using the almost trivial identity $d\theta^{\mu}{}_{\nu}=\frac{1}{2}\left(d\theta_{\alpha\beta}\right)\left(\mathbf{V}^{\alpha\beta}\right)^{\mu}{}_{\nu}$, they may be written in matrix form as

[TABLE]

Eq. (35) may be compared with the relation $\delta\gamma^{\mu}=\frac{1}{2}\left(d\theta_{\alpha\beta}\right)\left(\mathbf{V}^{\alpha\beta}\right)^{\mu}{}_{\nu}\gamma^{\nu}$, where $\gamma^{\mu}\equiv e^{\mu}{}_{\alpha}\gamma^{a}$, in the standard Dirac (vierbein)formalism. The ’extra’ commutator term in Eq. (35) is due to the fact that the row and column indices of $\mathbf{M}^{\mu}$ transform as well under Lorentz transformations, in compliance with Eq. (33). This is of course also the reason for the commutator term in Eq. (36). In conjunction with Eqs. (31)-(32), these relations then finally imply that

[TABLE]

These two relations are relevant for the proof of (local) Lorentz invariance of some specific action to be introduced in Sec. 4 below. From Eqs. (33)-(34) it follows that the Lorentz covariant derivatives of $M^{\mu\rho}{}_{\sigma}$ and $N_{i}{}^{\rho}{}_{\sigma}$ are necessarily given by

[TABLE]

the identically vanishing of which is due to Eqs. (3), (7)-(8), and (15)-(16). Using the almost trivial identity $\omega^{\rho}{}_{\sigma\mu}=\frac{1}{2}\left(\omega_{\alpha\beta\mu}\right)\left(\mathbf{V}^{\alpha\beta}\right)^{\rho}{}_{\sigma}$, they may be written in matrix form as

[TABLE]

where $\mathbf{\nabla}_{\tau}\mathbf{M}^{\mu}$ and $\mathbf{\nabla}_{\tau}\mathbf{N}_{i}$ (boldfaced nablas) mean, respectively, type $\left(2,2\right)$ and type $\left(1,2\right)$ world tensors with components $\left(\mathbf{\nabla}_{\tau}\mathbf{M}^{\mu}\right)^{\rho}{}_{\sigma}\equiv\nabla_{\tau}M^{\mu\rho}{}_{\sigma}$ and $\left(\mathbf{\nabla}_{\tau}\mathbf{N}_{i}\right)^{\rho}{}_{\sigma}\equiv\nabla_{\tau}N_{i}{}^{\rho}{}_{\sigma}$. [Notational remark: A boldfaced $\nabla$ and/or $D$ will be used whenever the covariant derivative acts on a matrix/vector-valued quantity, to remind the reader that the Levi-Civita covariant derivative will have to act also on the hidden row and/or column world indices, thus producing one or two extra Christoffel terms when fully expanded in tensor components. For the case just given, $\nabla_{\tau}\mathbf{M}^{\mu}$ and $\nabla_{\tau}\mathbf{N}_{i}$ (no boldface) could be mistaken to mean $\nabla_{\tau}\mathbf{M}^{\mu}=\partial_{\tau}\mathbf{M}^{\mu}+\Gamma^{\mu}{}_{\nu\tau}\mathbf{M}^{\nu}\neq\mathbf{\nabla}_{\tau}\mathbf{M}^{\mu}$ and $\nabla_{\tau}\mathbf{N}_{i}=\partial_{\tau}\mathbf{N}_{i}\neq\mathbf{\nabla}_{\tau}\mathbf{N}_{i}$. End of remark.]

A final note: The vector and spinor representations $\mathbf{V}^{\mu\nu}$ and $\mathbf{S}^{\mu\nu}$, which both depend only on the metric, are both (locally) Lorentz invariant, i.e., $\delta\mathbf{V}^{\mu\nu}=\delta\mathbf{S}^{\mu\nu}=\mathbf{0}$. This is reassuring, as the opposite case, i.e., having Lorentz generators that were not Lorentz invariant, would be somewhat of a conceptual quagmire.

4 Action and Euler-Lagrange equations of motion

This is the main section of the paper in which the pieces laid out in the previous section (on preliminaries) come together.

Consider in global Minkowski spacetime, in Cartesian coordinates $x^{\mu}$, endowed with spacetime-independent $n^{\mu}$ and $\overline{n}^{\mu}$ for which $g_{\mu\nu}\equiv n_{\mu}n_{\nu}-\overline{n}_{\mu}\cdot\overline{n}_{\nu}=\eta_{\mu\nu}$, the following action:

[TABLE]

where $N_{\rho\sigma}=N_{\rho\sigma}\left(a\right)\equiv a^{i}N_{i\rho\sigma}$ for some constants $a^{i}\in\mathbb{R}$ obeying $a^{i}a_{i}=1$. Here, $M_{\mu\rho\sigma}$ and $N_{i\rho\sigma}$ are given by Eqs. (15)-(16). The action will be considered at the classical level using (complex) Grassmann-valued $\psi^{\rho}$. The most distinctive property of $\mathcal{L}_{free}^{W}$ is that the spinor field carries a world index (the letter $W$ referring to this), as advertised in the Introduction, rather than a standard (Lorentz) spinor index. Using Eqs. (13)-(14), the Lagrangian may also be written in matrix notation as

[TABLE]

where $\mathbf{M}^{\mu}$ and $\mathbf{N}=a^{i}\mathbf{N}_{i}$ are determined by Eqs. (17)-(18), and where $\mathbf{\psi}$ is the four-column vector with components $\left(\mathbf{\psi}\right)^{\mu}=\psi^{\mu}$, obviously. The hat-hermiticity of $\mathbf{M}^{\mu}$, Eq. (22), guarantees that the kinetic part of the Lagrangian is complex self-conjugate; and the hat-antisymmetry of $\mathbf{N}$, Eq. (23), guarantees that the Majorana-like mass term is both complex self-conjugate and nontrivial. Eqs. (37)-(38) guarantee that the Lagrangian is globally invariant under the following Lorentz transformation (of the fundamental fields):

[TABLE]

[TABLE]

using the spacetime-independency of $M^{\mu}{}_{\rho\sigma}$ and $N_{\rho\sigma}$, due to the assumed spacetime-independency of $n^{\mu}$ and $\overline{n}^{\mu}$; or, equivalently, in matrix notation (by raising the $\rho$-index):

[TABLE]

As they should be, these Euler-Lagrange equations of motion are Klein-Gordon compatible, i.e., any plane wave solution is on mass shell, because

[TABLE]

using 1.) $\mathbf{E}_{free}^{W}$ in the form $\mathbf{E}_{free}^{W}=\left(\mathrm{i}\mathbf{M}^{\mu}K\partial_{\mu}-m\mathbf{N}\right)K\mathbf{\psi}$ with $K$ the operator of complex conjugation, 2.) Eqs. (19)-(20), and 3.) $\mathbf{N}^{2}=a^{i}a^{j}\mathbf{N}_{i}\mathbf{N}_{j}=\mathbf{1}$ due to Eq. (21) and the constraint $a^{i}a_{i}=1$, compare previous.

Now, in analogy with the standard procedure for switching on gravitational and/or inertial forces, compare the Introduction, 1.) let the coordinates be arbitrary, 2.) let $n^{\mu}$ and $\overline{n}^{\mu}$ be subject only to the orthonormality conditions given by Eq. (2), the metric $g_{\mu\nu}$ itself thus becoming arbitrary, and 3.) perform the substitution

[TABLE]

with $\omega_{\alpha\beta\mu}$ determined by Eq. (9), and $\mathbf{S}^{\alpha\beta}$ determined by Eq. (24). Note that as the spinor field now carries a world index, the explicit appearence of $\nabla_{\mu}$ is mandatory, in contrast to Eq. (1), where $\psi$ carries only a spinor index. In matrix notation, Eq. (44) may be written as

[TABLE]

where $\mathbf{\nabla}_{\mu}\mathbf{\psi}$ means a type $\left(1,1\right)$ world tensor field with components $\left(\mathbf{\nabla}_{\mu}\mathbf{\psi}\right)^{\rho}=\nabla_{\mu}\psi^{\rho}$, compare previous remark concerning boldfaced nabla. As shown in the Appendix at the end of this paper, $\mathbf{D}_{\mu}\mathbf{\psi}$ is a proper Lorentz covariant derivative in the sense that it transforms as $\delta\left(\mathbf{D}_{\mu}\mathbf{\psi}\right)=\frac{1}{2}\left(d\theta_{\alpha\beta}\right)\mathbf{S}^{\alpha\beta}\mathbf{D}_{\mu}\mathbf{\psi}$ under Eq. (42) with $d\theta_{a\beta}$ arbitrary. The action $S_{free}^{W}$ above then becomes the following coordinate invariant and locally Lorentz invariant action:

[TABLE]

where now $M^{\mu}{}_{\rho\sigma}$ and $N_{\rho\sigma}$ are generally spacetime-dependent. Explicitly expanding the covariant derivatives, the Lagrangian is given by

[TABLE]

from which it readily follows that

[TABLE]

from which in turn it follows that the Euler-Lagrange equations of motion for $\psi$ are given by

[TABLE]

using the identity $\partial_{\mu}\sqrt{-g}=\sqrt{-g}\Gamma^{\tau}{}_{\tau\mu}$. Expressing $\partial_{\mu}M^{\mu}{}_{\rho\sigma}$ in terms of $\nabla_{\mu}M^{\mu}{}_{\rho\sigma}$ (and three Christoffel terms), these equations may be rewritten as

[TABLE]

Being now manifestly tensorial, there being no explicit Christoffel symbols present, these equations can be rewritten, by unproblematically raising/lowering various indices, in matrix form as follows (note boldfaced nabla, compare previous remark):

[TABLE]

using the identity $\left(\mathbf{S}^{\alpha\beta\ast}\right)_{\rho}{}^{\tau}=\left(\mathbf{S}^{\alpha\beta\hat{\dagger}}\right)^{\tau}{}_{\rho}$, and Eqs. (31) and (39). But $\mathbf{E}_{grav}^{W}$, as thus defined, is simply $\mathbf{E}_{free}^{W}$, Eq. (43), subjected to the substitution Eq. (45), in conjunction with letting $\mathbf{M}^{\mu}$ and $\mathbf{N}$ become generically spacetime-dependent. Therefore, in the present formalism, the Lorentz gauging procedure commute with the Euler-Lagrange variational procedure, as asserted in the Introduction.

Conclusion

The formalism, as presented in this paper, for the coupling of a spinor field to the gravitational field has the following two distinguishing properties:

• •

Commutativity of the Lorentz gauging procedure and the Euler-Lagrange variational procedure, respectively, compare Sec. 4. In contrast, compare the Introduction, this property is not satisfied by the standard vierbein formalism for the coupling of a Dirac fermion to the gravitational field, the equivalence principle thus seemingly being here violated, quite unsettlingly.

• •

The use of world indices only, there being present neither Lorentz vector indices nor Lorentz spinor indices. Although this property is regarded by the author to be much less important than the commutativity property, previous bullet item, it nonetheless endows the formalism with the following two attractive properties: 1.) concerning indices, it is more parsimonious than the standard formalism is, and 2.) it seems to treat all types of fields, tensorial and spinorial ones, on an equal footing, contrary to the standard formalism.

There remains of course much to be investigated. Possible subjects are the following ones, just to mention some:

• •

Internal symmetry of the Lagrangian, and the coupling of $\mathbf{\psi}$ to the electromagnetic field.

• •

Discrete symmetries: $C$, $P$, and $T$.

• •

Free-particle solutions of the equations of motion for $\mathbf{\psi}$.

• •

Hamiltonian and the possible quantization of the theory.

The author intends to return to some or all of these issues in one or more future papers while at the same time of course most warmly welcoming anyone interested in contributing.

5 Appendix

The derivative $\mathbf{D}_{\mu}\mathbf{\psi}$, as defined by Eq. (45), is a proper Lorentz covariant derivative in the sense that it transforms as $\delta\left(\mathbf{D}_{\mu}\mathbf{\psi}\right)=\frac{1}{2}\left(d\theta_{\alpha\beta}\right)\mathbf{S}^{\alpha\beta}\mathbf{D}_{\mu}\mathbf{\psi}$ under Eq. (42) with $d\theta_{a\beta}$ arbitrary. Proof: The left-hand side is given by (note various boldface nablas)

[TABLE]

using $\delta\nabla_{\mu}=0$ and $\delta\mathbf{S}^{\alpha\beta}=0$ due to $\delta g_{\mu\nu}=0$, and $\mathbf{\nabla}_{\rho}\mathbf{S}^{\alpha\beta}=\mathbf{0}$ (boldface nabla) due to $\nabla_{\rho}g_{\mu\nu}=0$; and the right-hand side is given by (note boldface nabla)

[TABLE]

Using Eq. (27), these two expressions will be equal if and only if

[TABLE]

which is in fact so due to Eq. (10). End of proof.

This proof is structurally analogous to the proof in the standard formalism for the coupling of a Dirac spinor field to the gravitational field, compare again [1, Sec. 31.A] and [2, Sec. 12.1], that the Lorentz covariant derivative of a Dirac spinor field transforms properly under local Lorentz transformations. It is given here, nonetheless, for the benefit of the reader.