The symmetry group $GL(4, \mathbf{R})$

We have seen in Section 3.2 that, under some conditions, the cone $\mathcal{T}^+$ is completely determined by the value of the fundamental length $\ell$. Following refs. [7,11,12,14], we now describe the same cone in a different way, which permits an easier discussion of its properties.

We use the Dirac matrices $\gamma_i$ (see Section 0.3) in a Majorana representation. They are real and satisfy the conditions

$$\gamma_i^T = - C \gamma_i C^{-1}, \qquad \det \gamma_i = 1,$$ (3.15)

where $\gamma_i^T$ is the transposed matrix and $C$ is a real antisymmetric matrix, determined by this equation up to a numeric factor. It is determined up to the sign if we require that its Pfaffian is equal to $-1$, namely that

$$2^{-3} \epsilon^{ABCD} C_{AB} C_{CD} = -1,$$ (3.16)

a choice that is useful for the following developments and implies some restriction to the representation adopted for the $\gamma$-matrices. This condition can be rewritten in any of the following equivalent useful forms

$$2^{-3} \epsilon_{ABCD} (C^{-1})^{AB} (C^{-1})^{CD} = -1,$$ (3.17)

$$\epsilon_{ABCD} = - C_{AB} C_{CD} - C_{AC} C_{DB} - C_{AD} C_{BC},$$

$$\epsilon^{ABCD} = - (C^{-1})^{AB} (C^{-1})^{CD}$$

$$- (C^{-1})^{AC} (C^{-1})^{DB} - (C^{-1})^{AD} (C^{-1})^{BC}$$ (3.18)

and it implies that

$$\det C = 1.$$ (3.19)

In a suitable representation one can put $C = \gamma_0$, but one has to remember that these matrices have a different spinor nature as it is clear if we introduce the spinor covariant and contravariant indices and write $\gamma_i^A{}_B$, $C_{AB}$, $(C^{-1})^{AB}$. In any case, we choose the sign of $C$ in such a way that $C \gamma^0$ is a positive definite matrix.

We define the real symmetric $4 \times 4$ matrices

$$\Gamma_i = \ell^{-1} \gamma_i C^{-1}, \qquad \Gamma_{[ik]} = 2^{-1} (\gamma_{i} \gamma_{k} - \gamma_{k} \gamma_{i}) C^{-1}.$$ (3.20)

and we put

$$b = b^{\alpha} \Gamma_{\alpha}.$$ (3.21)

We also define the real symmetric matrices

$$\breve\Gamma^{\alpha} = (\Gamma_{\alpha})^{-1}, \qquad \bre... ... = - 2^{-1} C (\gamma^{i} \gamma^{k} - \gamma^{k} \gamma^{i}),$$ (3.22)

and from the property

$$2^{-2} \mathrm{Tr}\,(\breve\Gamma^{\alpha} \Gamma_{\beta}) = \delta^{\alpha}_{\beta},$$ (3.23)

we obtain the inverse formula

$$b^{\alpha} = 2^{-2} \mathrm{Tr}\,(\breve\Gamma^{\alpha} b).$$ (3.24)

We define $\mathcal{T}^+$ by requiring that the real symmetric matrix $b$ is positive semidefinite, namely that

$$\psi^T b \psi \geq 0$$ (3.25)

for any choice of the real spinor $\psi$. It is clear that $\mathcal{T}^+$ is symmetric under the transformations

$$b \to a b a^T, \qquad a \in GL(4, \mathbf{R}).$$ (3.26)

Note that $a$ and $- a$ give rise to the same transformation of $\mathcal{T}$. This transformation property means that the matrix $b^{AB}$ represents a contravariant symmetric $GL(4, \mathbf{R})$ spinor. The possible physical relevance of many-dimensional spaces with a local symmetry group different from a many-dimensional Lorentz group has been examined, in a different context, in ref. [85]

Since a complex $2 \times 2$ matrix can be considered as a real $4 \times 4$ matrix, the 16-dimensional symmetry group $GL(4, \mathbf{R})$ contains a subgroup isomorphic to $SL(2, \mathbf{C})$, namely to the universal covering of the proper orthochronous Lorentz group. The infinitesimal Lorentz transformations are given by

$$a \sim 1 + 2^{-1} \zeta^{[ik]} \Sigma_{[ik]},$$ (3.27)

where the matrices $\Sigma_{[ik]}$ are given by eq. (1.17). From this formula we obtain, as it was expected,

$$\delta b^i = \zeta^i{}_j b^j, \qquad \delta b^{[ik]} = \zeta^i{}_j b^{[jk]} + \zeta^k{}_j b^{[ij]}.$$ (3.28)

Since $\mathcal{T}^+$ is a Lorentz invariant closed cone with nonempty interior, it coincides with the cone defined in Section 3.2 and we have only to show that the the parameter $\ell$ is the same. It is sufficient to remark that the extremal elements defined by eq. (3.12) with $a = \ell^{-1}$ belong to the boundary of $\mathcal{T}^+$ and on this boundary the determinant $\det b$ vanishes.

By means of a direct calculation, using an explicit representation of the gamma matrices, we obtain

$$\det b = (\ell^{-2} (b^0)^2 - \ell^{-2} \Vert\mathbf{b}\Vert^2 - \Ve... ...\mathbf{b}''\Vert^2 )^2 - 4 \ell^{-2} \Vert\mathbf{b} \times \mathbf{b}'\Vert^2$$

$$- 4 \Vert\mathbf{b}' \times \mathbf{b}''\Vert^2 - 4 \ell^{-2} \Vert\... ...}\Vert^2 + 8 \ell^{-2} b^0 \, \mathbf{b} \cdot \mathbf{b}' \times \mathbf{b}''.$$ (3.29)

A simple substitution shows that the required condition is satisfied, if the parameter $\ell$ is the same as the one defined in Section 3.2.

It may be stimulating to propose a physical interpretation of $\det b$ [22]. We consider a moving frame $\tau \to s(\tau)$ with the property

$$\frac{ds(\tau)}{d \tau} = b^{\alpha} A_{\alpha} \in \mathcal{T}^+$$ (3.30)

and we consider the integral

$$\ell \int_{\tau_0}^{\tau} (\det b)^{1/4} \, d \tau.$$ (3.31)

For $\ell \to 0$ this integral takes the form

$$\int_{\tau_0}^{\tau} ((b^0)^2 - \Vert\mathbf{b}\Vert^2)^{1/2} \, d \tau,$$ (3.32)

which gives the relativistic formula for the time measured by an ideal clock moving with the frame $s(\tau)$ and takes into account the influence of the velocity on the clock rate. It is natural to assume that eq. (3.31) also takes into account the influence of the acceleration and the angular velocity of the frame and describes an ideal accelerated clock.

It is clear that all the real clocks are influenced by the inertial forces due to acceleration and by the centrifugal forces due to rotation. These forces can even destroy the clock mechanism. Eq. (3.31) could describe a dependence that cannot be made arbitrarily small by building more and more robust mechanisms and cannot be explained in terms of the normal theory.

The function $d(B) = (\det b)^{1/4}$ is homogeneous of degree one and can be considered as a pseudo-norm defined on the cone $T^+$. We choose a representation of the $\gamma$ matrices in which $C \gamma^0 = 1$ and we indicate by $\beta_1,\ldots, \beta_4$ the positive eigenvalues of $b$. Then we have

$$d(B) = (\beta_1 \beta_2 \beta_3 \beta_4)^{1/4} \leq 2^{-2} (... ...a_2 + \beta_3 + \beta_4) = 2^{-2} \mathrm{Tr}\,(b) = \ell b^0.$$ (3.33)

We can also deduce the interestig equation

$$d(B + B') \geq d(B) + d(B').$$ (3.34)

Note the different inequality sign with respect to the familiar "triangular" property of a norm. Since the determinants are invariant under $SL(4, \mathbf{R})$, we can apply a transformation of this group in such a way that $b + b'$ is a multiple of the unit matrix. It follws that

$$d(B + B') = 2^{-2} \mathrm{Tr}\,(b + b') = 2^{-2} (\mathrm{Tr}\,b + \mathrm{Tr}\,b') \geq d(B) + d(B').$$ (3.35)

There is an analogy with the pseudo-norm of a relativistic 4-vector, defined on the future cone by the formula

$$d(v) = (- v^i v_i)^{1/2}.$$ (3.36)

It satisfies an inequality similar to eq. (3.34). In the tangent spaces of the spacetime manifold, it is given by the square root of a quadratic form defined by the metric tensor. If the quadratic form were definite positive, $d(v)$ would be a norm and $\mathcal{M}$ would be a Riemannian space, otherwise it is a pseudo-Riemannian space. In the space $\mathcal{S}$ the pseudo-norm is not the square root of a quadratic form, but the fourth root of a form of degree 4. If $d(B)$ were a norm, positive and defined for every vector $B$, $\mathcal{S}$ would be a Finslerian space [86]. Since this is not the case, it is a pseudo-Finslerian space.

It is interesting to remark the double interplay between symmetry groups and cones [21]. The rotational symmetry together with a choice of a maximal valocity determines the future cone in a tangent space of $\mathcal{M}$. The symmetry group of this cone is the product of the Lorentz group and the dilatation group. The Lorentz symmetry together with a choice of a maximal acceleration (or of a fundamental length) determines the cone $\mathcal{T}^+$. The symmetry group of this cone is $GL(4, \mathbf{R})$.