*2.1. Geometry and Its Matter Content*

To build the geometric background on which to define kinematic fields, we start with the symmetry principle at the basis of any theory in physics: covariance under the most general transformation of coordinates. We will see in what way from such a general environment a natural definition of matter field will spontaneously emerge.

### 2.1.1. Tensor and Gauge Fields

The principle of covariance under the most general transformation of coordinates is possible one of the most self-evident principles in all of physics: it states that our way of writing equations might a priori be conditioned in principle by the coordinates we choose, but observable properties should not feel affected by coordinate artifacts brought by us. This means that of all possible manners we have to write physics, there must be one that is not influenced by any choice of coordinates, or in other words, this specific way of writing physics has to be invariant between different coordinate systems.

To see how this is possible, we start with the following definition. Suppose that a certain physical quantity can be described in terms of the object *T α*...*σ ρ*...*ζ* such that it is written as *T α*...*σ ρ*...*ζ* (*x*) with respect to coordinates *x*, and it is written as *T* ′*α*...*σ ρ*...*ζ* (*x* ′ ) with respect to coordinates *x* ′ in general, and suppose that

$$T^{\prime \alpha \dots \sigma}\_{\rho \dots \zeta} = \frac{\partial \mathbf{x}^{\beta}}{\partial \mathbf{x}^{\prime \rho}} \dots \frac{\partial \mathbf{x}^{\rho}}{\partial \mathbf{x}^{\prime \zeta}} \frac{\partial \mathbf{x}^{\prime \alpha}}{\partial \mathbf{x}^{\nu}} \dots \frac{\partial \mathbf{x}^{\prime \sigma}}{\partial \mathbf{x}^{\tau}} T^{\nu \dots \tau}\_{\beta \dots \theta}$$

where *x* ′ =*x* ′ (*x*) determines the passage from the first to the second system of coordinates. If this happens, such a quantity is called a tensor. Now, suppose that one specific property of this quantity be described as

$$T^{\upsilon\dots\tau}\_{\beta\dots\theta} = 0$$

in the first system of coordinates. According to the above definition, then, we have that

$$T^{\prime \alpha \dots \sigma}\_{\rho \dots \zeta} = \frac{\partial x^{\beta}}{\partial x^{\prime \rho}} \dots \frac{\partial x^{\theta}}{\partial x^{\prime \xi}} \frac{\partial x^{\prime \alpha}}{\partial x^{\prime \tau}} \dots \frac{\partial x^{\prime \tau}}{\partial x^{\tau}} T^{\prime \dots \tau}\_{\beta \dots \theta} = 0$$
 
$$= \frac{\partial x^{\beta}}{\partial x^{\prime \rho}} \dots \frac{\partial x^{\theta}}{\partial x^{\prime \xi}} \frac{\partial x^{\prime \alpha}}{\partial x^{\prime \tau}} \dots \frac{\partial x^{\prime \tau}}{\partial x^{\tau}} 0 \equiv 0$$

and thus

$$T^{\prime a\dots\sigma}\_{\rho\dots\zeta} = 0$$

showing that the same property pertains to that quantity also in the second system of coordinates as well. In this way, we have that, if the property of a quantity is encoded as the vanishing of a tensor, then we can be certain that such a property pertains to that quantity regardless of the system of coordinates. In addition, this is just covariance.

The principle of covariance is therefore implemented in the geometry by the straightforward requirement that this geometry be written in terms of tensors. Therefore, let be given two systems of coordinates as *x* and *x* ′ related by the most general coordinate transformation *x* ′ =*x* ′ (*x*) and a set of functions of these coordinates written with respect to the first and the second system of coordinates as *T*(*x*) and *T* ′ (*x* ′ ) and such that, for a coordinate transformation, they are related by

$$T^{\prime a\dots \sigma}\_{\rho\dots\zeta} = \text{sign } \det \left( \frac{\partial \mathbf{x}^{\prime}}{\partial \mathbf{x}} \right) \frac{\partial \mathbf{x}^{\theta}}{\partial \mathbf{x}^{\prime \theta}} \dots \frac{\partial \mathbf{x}^{\theta}}{\partial \mathbf{x}^{\prime \zeta}} \frac{\partial \mathbf{x}^{\prime \mu}}{\partial \mathbf{x}^{\nu}} \dots \frac{\partial \mathbf{x}^{\prime \sigma}}{\partial \mathbf{x}^{\tau}} T^{\nu}\_{\beta\dots\theta} \tag{1}$$

Then, this quantity is called *tensor* or *pseudo-tensor*, according to whether the sign of the determinant of such a transformation is positive or negative. For a tensor with at least two upper or two lower indices, we might switch the two indices obtaining a tensor called *transposition* of the original tensor in those two indices, and if it happens to be equal to the initial tensor up to the sign plus or minus, we say that the tensor is *symmetric* or *antisymmetric* in those two indices, respectively. Given a tensor with at least one upper and one lower index, we can consider one of the upper and one of the lower indices forcing them to have the same value and performing the sum over every possible value of those indices obtaining a tensor called *contraction* in those indices, and this process can be repeated until we reach a tensor whose contraction is zero, called *irreducible*. Particular cases are tensors having one index called *vectors*, while tensors without any index are called *scalars*. Tensors with the same index configuration can be *summed* and any two tensors can be *multiplied* in a component-by-component way, according to the usual rules of algebraic calculus as they are well known.

There is therefore no need to spend more time in the algebraic properties of tensors. However, differential properties of tensors need some deepening. The problem with differentiation applied to the case of tensors is that such an operation spoils the transformation law of a tensor in very general circumstances. Thus, if we want to construct an operation that is able to generalize the usual derivative up to a derivative that respects covariance, we must begin by noticing that, because a tensor is a set of fields, in general, it will have two types of variations: the first is due to the fact that tensors fields are fields, coordinates dependent, and so a local structure must be present as

$$\begin{aligned} \partial\_{\text{local}} \Delta T^{\mathfrak{a}\_1 \dots \mathfrak{a}\_j}\_{\beta\_1 \dots \beta\_i} &= T^{\mathfrak{a}\_1 \dots \mathfrak{a}\_j}\_{\beta\_1 \dots \beta\_i} (\mathfrak{x}') - T^{\mathfrak{a}\_1 \dots \mathfrak{a}\_j}\_{\beta\_1 \dots \beta\_i} (\mathfrak{x}) = \\ &= \partial\_{\mu} T^{\mathfrak{a}\_1 \dots \mathfrak{a}\_j}\_{\beta\_1 \dots \beta\_i} (\mathfrak{x}) \delta \mathfrak{x}^{\mu} \end{aligned}$$

at the first order infinitesimal; the second is due to the fact that tensors fields are tensors, so a system of components, and thus a re-shuffling of the different components must be allowed according to

$$\begin{split} \text{structure} \Delta T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} &= T^{\prime \alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} - T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} = \\ &= [ (\delta \Gamma^{\alpha\_{1}}\_{\theta} T^{\theta\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + ... + \delta \Gamma^{\alpha\_{j}}\_{\theta} T^{\alpha\_{1}\ldots\theta}\_{\beta\_{1}\ldots\beta\_{i}}) - \\ &- (\delta \Gamma^{\theta}\_{\beta\_{1}} T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\theta\ldots\beta\_{i}} + ... + \delta \Gamma^{\theta}\_{\beta\_{i}} T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\theta}) ) \end{split}$$

as the most general form in which this can be done while respecting the fact that the differential structure requires the linearity and the Leibniz rule, and again at the first order of infinitesimal. In full, we have

$$\begin{split} \Delta T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} &= \operatorname{local} \Delta T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \operatorname{structure} \Delta T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} = \delta \\ &= \partial\_{\mu} T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}}(\mathbf{x}) \delta \mathbf{x}^{\mu} + \\ &+ [ (\delta \Gamma^{\alpha\_{1}}\_{\theta} T^{\theta\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \ldots + \delta \Gamma^{\alpha\_{i}}\_{\theta} T^{\alpha\_{1}\ldots\theta}\_{\beta\_{1}\ldots\beta\_{i}}) - \\ &- (\delta \Gamma^{\theta}\_{\beta\_{1}} T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\theta\ldots\theta\_{i}} + \ldots + \delta \Gamma^{\theta}\_{\beta\_{i}} T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\theta}) ] \end{split}$$

at the first order infinitesimal. Thus, defining *δ*Γ *α <sup>β</sup>* =Γ *α βµδx <sup>µ</sup>* and dividing by *δx µ* , we obtain that

$$\begin{aligned} D\_{\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} &= \partial\_{\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \\ + (\Gamma^{\alpha\_{1}}\_{\theta\mu}T^{\theta\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \ldots + \Gamma^{\alpha\_{j}}\_{\theta\mu}T^{\alpha\_{1}\ldots\theta}\_{\beta\_{1}\ldots\beta\_{i}}) - \\ - (\Gamma^{\theta}\_{\beta\_{1}\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\theta\ldots\beta\_{i}} + \ldots + \Gamma^{\theta}\_{\beta\_{i}\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\theta}), \end{aligned}$$

after taking the limit. This is the most general form of potential covariant derivative. To see that this derivative is indeed covariant, we have to require that Γ *α βµ* transforms with a specific non-tensorial transformation law such as to compensate for the non-tensorial transformation law of the partial derivative. In the simplest case of one tensorial index, we have that the derivative is

$$D\_l V^\alpha = \partial\_l V^\alpha + V^\beta \Gamma^\alpha\_{\beta \nu}$$

whose transformation law is given by

*∂x β ∂x* ′*β* ′ *∂x* ′*α* ′ *∂x <sup>α</sup>* (*∂βV <sup>α</sup>* + *V ρ*Γ *α ρβ*) = *<sup>∂</sup><sup>x</sup> β ∂x* ′*β* ′ *∂x* ′*α* ′ *∂x <sup>α</sup> DβV <sup>α</sup>* = = (*DβV α* ) ′ = (*∂βV <sup>α</sup>*+*V ρ*Γ *α ρβ*) ′ =*∂<sup>β</sup>* ′*V α* ′ +*V ρ* ′ Γ ′*α* ′ *ρ* ′*β* ′ = = *<sup>∂</sup><sup>x</sup> θ ∂x* ′*β* ′ *∂ ∂x θ ∂x* ′*α* ′ *∂x <sup>α</sup> V α* + *<sup>∂</sup><sup>x</sup>* ′*ρ* ′ *∂x <sup>ρ</sup> V ρ*Γ ′*α* ′ *ρ* ′*β* ′ = = *<sup>∂</sup><sup>x</sup> θ ∂x* ′*β* ′ *∂x* ′*α* ′ *∂x α ∂V α ∂x <sup>θ</sup>* + *<sup>∂</sup><sup>x</sup> θ ∂x* ′*β* ′ *∂ ∂x θ ∂x* ′*α* ′ *∂x <sup>α</sup> V <sup>α</sup>* + *<sup>∂</sup><sup>x</sup>* ′*ρ* ′ *∂x <sup>ρ</sup> V ρ*Γ ′*α* ′ *ρ* ′*β* ′

in which terms with the derivatives disappear. Then,

$$\frac{\partial \boldsymbol{x}^{\boldsymbol{\beta}}}{\partial \boldsymbol{\alpha}^{\prime \boldsymbol{\beta}^{\prime}}} \frac{\partial \boldsymbol{x}^{\prime \boldsymbol{\alpha}^{\prime}}}{\partial \boldsymbol{\alpha}^{\prime \boldsymbol{\alpha}}} V^{\rho} \Gamma^{\alpha}\_{\rho \boldsymbol{\beta}} = \frac{\partial \boldsymbol{x}^{\theta}}{\partial \boldsymbol{x}^{\prime \boldsymbol{\beta}^{\prime}}} \frac{\partial}{\partial \boldsymbol{x}^{\theta}} \frac{\partial \boldsymbol{x}^{\prime \boldsymbol{\alpha}^{\prime}}}{\partial \boldsymbol{x}^{\boldsymbol{\alpha}}} V^{\boldsymbol{\alpha}} + \frac{\partial \boldsymbol{x}^{\prime \rho^{\prime}}}{\partial \boldsymbol{x}^{\rho}} V^{\rho} \Gamma^{\prime \alpha^{\prime}}\_{\rho^{\prime} \boldsymbol{\beta}^{\prime}}$$

and since this has to hold for any vector

$$\frac{\partial \boldsymbol{x}^{\beta}}{\partial \boldsymbol{x}^{\prime \beta^{\prime}}} \frac{\partial \boldsymbol{x}^{\prime \boldsymbol{\alpha}^{\prime}}}{\partial \boldsymbol{x}^{\alpha}} \boldsymbol{\Gamma}^{\alpha}\_{\rho \beta} = \frac{\partial \boldsymbol{x}^{\theta}}{\partial \boldsymbol{x}^{\prime \beta^{\prime}}} \frac{\partial}{\partial \boldsymbol{x}^{\theta}} \frac{\partial \boldsymbol{x}^{\prime \boldsymbol{\alpha}^{\prime}}}{\partial \boldsymbol{x}^{\theta}} + \frac{\partial \boldsymbol{x}^{\prime \rho^{\prime}}}{\partial \boldsymbol{x}^{\rho}} \boldsymbol{\Gamma}^{\prime \boldsymbol{\alpha}^{\prime}}\_{\rho^{\prime} \beta^{\prime}}$$

which is the non-tensorial transformation that the set of coefficients Γ *α ρβ* must have to ensure that the full derivative transforms as a tensor in this very specific case with a vector field. However, quite remarkably, the very same non-tensorial transformation of Γ *α ρβ* can be used for each term in the most general form of derivative for a generic tensor, and so the obtained result is completely general.

The set of coefficients Γ *α ρβ* have no specific symmetry properties in the lower indices, and consequently we have that we can write

$$\Gamma^{\alpha}\_{\mu\nu} \equiv \frac{1}{2} (\Gamma^{\alpha}\_{\mu\nu} + \Gamma^{\alpha}\_{\nu\mu}) + \frac{1}{2} (\Gamma^{\alpha}\_{\mu\nu} - \Gamma^{\alpha}\_{\nu\mu})^2$$

where the transformation properties of the full object is inherited by the first part, which is symmetric in the two lower indices, and it can be indicated as

$$
\Lambda^{\alpha}\_{\mu\nu} = \frac{1}{2} (\Gamma^{\alpha}\_{\mu\nu} + \Gamma^{\alpha}\_{\nu\mu}),
$$

while the second part

$$Q^{\alpha}\_{\;\;\;\mu\nu} = \Gamma^{\alpha}\_{\;\;\mu\nu} - \Gamma^{\alpha}\_{\;\;\nu\mu}$$

transforms as a tensor such that *Q<sup>α</sup> µν* <sup>=</sup> <sup>−</sup>*Q<sup>α</sup> νµ*, meaning that it is antisymmetric in its second pair of indices. Thus,

$$\Gamma^{\alpha}\_{\mu\nu} = \Lambda^{\alpha}\_{\mu\nu} + \frac{1}{2}Q^{\alpha}\_{\ \mu\nu}$$

in the most general case. As in the covariant derivatives, the connection enters linearly, and the splitting in symmetric and antisymmetric parts sums up to a linear combination of the tensor *Q<sup>α</sup> µν* plus the terms linear in the symmetric connection, which therefore forms yet another type of covariant derivative that is defined according to

$$\begin{split} \nabla\_{\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} &= \partial\_{\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \\ + (\Lambda^{\alpha\_{1}}\_{\theta\mu}T^{\theta\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \ldots + \Lambda^{\alpha\_{j}}\_{\theta\mu}T^{\alpha\_{1}\ldots\theta}\_{\beta\_{1}\ldots\beta\_{i}}) - \\ - (\Lambda^{\theta}\_{\beta\_{1}\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\theta\ldots\beta\_{i}} + \ldots + \Lambda^{\theta}\_{\beta\_{i}\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\theta}) \end{split}$$

and in it the fact that the symmetric connection is indeed symmetric allows for particularly simplified expressions in some special cases. For instance, taking the symmetric covariant derivative of a tensor with all lower indices gives

$$\nabla\_{\mu}T\_{\beta\_1\ldots\beta\_i} = \partial\_{\mu}T\_{\beta\_1\ldots\beta\_i} - \Lambda^{\theta}\_{\beta\_1\mu}T\_{\theta\ldots\beta\_i} - \ldots - \Lambda^{\theta}\_{\beta\_i\mu}T\_{\beta\_1\ldots\beta\_i}$$

which is particularly interesting because we see that the symmetric connection always saturates the same index in the upper position, so that, if we further specialize onto the case in which the tensor is completely antisymmetric, we obtain that

$$\begin{split} \nabla\_{[\mu}T\_{\beta\ldots\rho]} &= \nabla\_{\mu}T\_{\beta\ldots\rho} - \nabla\_{\beta}T\_{\mu\ldots\rho} + \ldots - \nabla\_{\rho}T\_{\beta\ldots\mu} = \Phi\\ &= \partial\_{\mu}T\_{\beta\ldots\rho} - \Lambda^{\sigma}\_{\beta\mu}T\_{\sigma\ldots\rho} - \ldots - \Lambda^{\sigma}\_{\rho\mu}T\_{\beta\ldots\sigma} - \\ &- \partial\_{\beta}T\_{\mu\ldots\rho} + \Lambda^{\sigma}\_{\mu\beta}T\_{\sigma\ldots\rho} + \ldots + \Lambda^{\sigma}\_{\rho\beta}T\_{\mu\ldots\sigma} + \ldots\\ &\ldots - \partial\_{\rho}T\_{\beta\ldots\mu} + \Lambda^{\sigma}\_{\beta\rho}T\_{\sigma\ldots\mu} + \ldots + \Lambda^{\sigma}\_{\mu\rho}T\_{\beta\ldots\sigma} = \\ &= \partial\_{\mu}T\_{\beta\ldots\rho} - \partial\_{\beta}T\_{\mu\ldots\rho} + \ldots - \partial\_{\rho}T\_{\beta\ldots\mu} = \partial\_{[\mu}T\_{\beta\ldots\rho]} \end{split}$$

where all symmetric connections cancelled off leaving an expression written only in terms of partial derivatives but that is a completely antisymmetric covariant derivative in the most general case. This is a very peculiar property of tensors having all lower indices and being completely antisymmetric in all of these indices, and there is an entire domain related to this type of tensors and covariant derivatives, in which tensors are known as forms and

the covariant derivatives are part of what is known as exterior calculus. Nevertheless, we will not discuss it here because we do not want to introduce even further mathematical concepts and after all forms and exterior derivatives are nothing but a specific type of tensors. We encourage the interested readers to study this domain on their own.

Thus, to summarize what we have done, we have that the set of functions Γ *ρ αβ* transforming as

$$
\Gamma^{\prime \rho}\_{\sigma \tau} = \left( \Gamma^{\mu}\_{\mu \nu} - \frac{\partial \mathbf{x}^{a}}{\partial \mathbf{x}^{\prime \mu}} \frac{\partial^{2} \mathbf{x}^{\prime \kappa}}{\partial \mathbf{x}^{\prime \nu} \partial \mathbf{x}^{\prime \mu}} \right) \frac{\partial \mathbf{x}^{\prime \rho}}{\partial \mathbf{x}^{\prime \nu}} \frac{\partial \mathbf{x}^{\mu}}{\partial \mathbf{x}^{\prime \nu}} \frac{\partial \mathbf{x}^{\nu}}{\partial \mathbf{x}^{\prime \tau}} \tag{2}
$$

is called *connection*, and it can be decomposed as

$$
\Gamma^{\rho}\_{\;\;\alpha\beta} = \Lambda^{\rho}\_{\;\alpha\beta} + \frac{1}{2} \mathcal{Q}^{\rho}\_{\;\;\;\alpha\beta} \tag{3}
$$

where Λ *ρ αβ* is a set of functions transforming according to the law of a connection but which are symmetric in the two lower indices called *symmetric connection* and

$$Q^{\rho}\_{\ \ a\beta} = \Gamma^{\rho}\_{\ a\beta} - \Gamma^{\rho}\_{\beta\alpha} \tag{4}$$

which is a tensor antisymmetric in the two lower indices called *torsion tensor*. In terms of the connection, we may write the *covariant derivative* in the most general case as

$$\begin{split} D\_{\mu}T^{\mu\_{1}\ldots\mu\_{j}}\_{\beta\_{1}\ldots\beta\_{l}} &= \partial\_{\mu}T^{\mu\_{1}\ldots\mu\_{j}}\_{\beta\_{1}\ldots\beta\_{l}} + \sum\_{k=1}^{k=j} \Gamma^{\alpha\_{k}}\_{\sigma\mu} T^{\alpha\_{1}\ldots\sigma\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{l}} - \\ &- \sum\_{k=1}^{k=i} \Gamma^{\sigma}\_{\beta\_{k}\mu} T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\sigma\ldots\beta\_{i}} \end{split} \tag{5}$$

decomposing as

$$\begin{split} D\_{\mu}T^{\mu\_{1}\ldots\mu\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} &= \nabla\_{\mu}T^{\mu\_{1}\ldots\mu\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} + \frac{1}{2}\sum\_{k=1}^{k=j} Q^{a\_{k}}\_{\sigma\mu}T^{\alpha\_{1}\ldots\sigma\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}} - \\ &- \frac{1}{2}\sum\_{k=1}^{k=i} Q^{\sigma}\_{\beta\_{k}\mu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\sigma\ldots\beta\_{i}} \end{split} \tag{6}$$

with spurious terms linear in the torsion tensor and

$$\begin{split} \nabla\_{\mu}T^{\mu\_{1}\ldots\mu\_{j}}\_{\beta\_{1}\ldots\beta\_{l}} &= \partial\_{\mu}T^{\mu\_{1}\ldots\mu\_{j}}\_{\beta\_{1}\ldots\beta\_{l}} + \sum\_{k=1}^{k=j} \Lambda^{\underline{a}\_{k}}\_{\sigma\mu} T^{\underline{a}\_{1}\ldots\sigma\ldots\underline{a}\_{j}}\_{\beta\_{1}\ldots\beta\_{l}} - \\ &- \sum\_{k=1}^{k=i} \Lambda^{\sigma}\_{\beta\_{k}\mu} T^{\underline{a}\_{1}\ldots\underline{a}\_{j}}\_{\beta\_{1}\ldots\sigma\ldots\underline{a}\_{i}} \end{split} \tag{7}$$

which is the *covariant derivative calculated with respect to the symmetric connection*. If we apply the last definition to the particular case of tensors with all lower indices and being completely antisymmetric, we get

$$\nabla\_{\left[\nu\right.} T\_{\left.\mu\dots\sigma\right]} = \partial\_{\left[\nu\right.} T\_{\left.\mu\dots\sigma\right]} \equiv (\partial T)\_{\left.\nu\mu\dots\sigma\right.} \tag{8}$$

which is still a tensor and such that it is completely antisymmetric called a *covariant curl* of the tensor field. When in the covariant derivative of a tensor with at least one upper index we contract the index of derivation with an upper index of the tensor field, we get what is known as *covariant divergence* in that index of the tensor field.

When we have introduced the concept of tensor, it naturally emerged that, in the definition, two types of indices were present, upper and lower, reflecting the fact that a tensor could transform according to two type of transformations, direct and inverse. However, these two types of transformation are two different forms of the same transformation, and so one should expect that the two types of indices be two different arrangements of the same system of components. Thus, there should be no difference in content if we move a given index up or down at will.

What this implies is that it should be possible to move indices up and down without losing or adding anything to the information content: this can be done by considering the

Kronecker tensor *δ α <sup>ν</sup>* and postulating the existence of two tensors *gαν* and *g αν* in general. Then, we can define the operation of raising and lowering of tensorial indices by considering that *A <sup>π</sup>gπν* and *Aπg πν* are tensors that are related to the initial ones but with the index lowered and raised, respectively, and so we may define these two tensors as *Aπg πν* <sup>≡</sup> *<sup>A</sup> ν* and *A <sup>π</sup>gπν* <sup>≡</sup> *<sup>A</sup><sup>ν</sup>* as the same tensors but with the index moved in a different position with respect to the initial one. While it is certainly useful to have the possibility to perform such an operation, we also have to consider that such an operation has a two-fold ambiguity concerning the fact that, besides the contractions *Aπg πν* <sup>≡</sup> *<sup>A</sup> <sup>ν</sup>* and *A <sup>π</sup>gπν* <sup>≡</sup> *<sup>A</sup>ν*, we may have the contractions *Aπg νπ* <sup>≡</sup> *<sup>A</sup> <sup>ν</sup>* and *A <sup>π</sup>gνπ* <sup>≡</sup> *<sup>A</sup><sup>ν</sup>* too. In addition, we may decide to raise the previously lowered index to the initial position or lower the previously raise index to the initial position, so that the above ambiguity becomes four-fold with *Aπg πνgσν* <sup>≡</sup> *<sup>A</sup><sup>σ</sup>* and *Aπg νπgσν* <sup>≡</sup> *<sup>A</sup><sup>σ</sup>* as well as *<sup>A</sup>π<sup>g</sup> πνgνσ* <sup>≡</sup> *<sup>A</sup><sup>σ</sup>* and *<sup>A</sup>π<sup>g</sup> νπgνσ* <sup>≡</sup> *<sup>A</sup><sup>σ</sup>* as equally good possibilities that may be considered. On the other hand, requiring that raising one index up and then lowering that index down give back the initial tensor in all of the four possibilities leads to the following relationships

$$\begin{aligned} A\_{\mu}(\operatorname{g}^{\mu\sigma}\mathcal{g}\_{\sigma\kappa}-\delta^{\mu}\_{\kappa})&=0 & A\_{\mu}(\operatorname{g}^{\sigma\mu}\mathcal{g}\_{\sigma\kappa}-\delta^{\mu}\_{\kappa})&=0\\ A\_{\mu}(\operatorname{g}^{\mu\sigma}\mathcal{g}\_{\kappa\sigma}-\delta^{\mu}\_{\kappa})&=0 & A\_{\mu}(\operatorname{g}^{\sigma\mu}\mathcal{g}\_{\kappa\sigma}-\delta^{\mu}\_{\kappa})&=0 \end{aligned}$$

for any possible tensor *Aµ*, so that

$$\begin{aligned} \left(g^{\mu\sigma}g\_{\sigma\kappa} - \delta^{\mu}\_{\kappa}\right) &= 0 \\ \left(g^{\mu\sigma}g\_{\kappa\sigma} - \delta^{\mu}\_{\kappa}\right) &= 0 \end{aligned} \qquad \begin{aligned} \left(g^{\sigma\mu}g\_{\sigma\kappa} - \delta^{\mu}\_{\kappa}\right) = 0 \\ \left(g^{\sigma\mu}g\_{\kappa\sigma} - \delta^{\mu}\_{\kappa}\right) &= 0 \end{aligned}$$

identically. Taking the differences

$$\begin{aligned} \mathcal{g}^{\mu\sigma}(\mathcal{g}\_{\sigma\kappa} - \mathcal{g}\_{\kappa\sigma}) &= 0 & (\mathcal{g}^{\sigma\mu} - \mathcal{g}^{\mu\sigma})\mathcal{g}\_{\sigma\kappa} &= 0 \\ \mathcal{g}^{\sigma\mu}(\mathcal{g}\_{\sigma\kappa} - \mathcal{g}\_{\kappa\sigma}) &= 0 & (\mathcal{g}^{\sigma\mu} - \mathcal{g}^{\mu\sigma})\mathcal{g}\_{\kappa\sigma} &= 0 \end{aligned}$$

we may work out that

$$\begin{aligned} \mathcal{g}\_{\mathfrak{A}\mathfrak{K}} &= \mathcal{g}\_{\mathfrak{K}\mathfrak{K}} \\ \mathcal{g}^{\mathfrak{A}\mathfrak{K}} &= \mathcal{g}^{\mathfrak{K}\mathfrak{K}} \end{aligned}$$

together with the condition

$$\mathcal{g}^{\sigma\mu}\mathcal{g}\_{\kappa\sigma} = \delta^\mu\_\kappa$$

meaning that, seen as matrices, they are symmetric and one the inverse of the other, and so, in particular, they are non-degenerate, as it has been demonstrated in [9]. This implies that what has been introduced to raise lower or lower upper indices has all the features of a metric and therefore these two tensors can also be identified with the metric of the space–time. We remark that this is exactly the opposite to the normal approach, where the metric is postulated, and then it is realized that it can be used to move up and down indices of tensors. The equivalence of these two a priori unrelated operations looks profound.

The metric determinant det(*gµν*) =*g* can never be zero, but it follows the transformation law

$$\mathbf{g'} = \det \left| \frac{\partial \mathbf{x}}{\partial \mathbf{x'}} \right|^2 \mathbf{g}$$

which is not the transformation law for a tensor. However, it can still be used to form a very important tensor as in the following. Consider in fact the non-tensorial quantity that is given by *ǫi*<sup>1</sup> *i*2*i*3*i*4 such that it is equal to the unity for an even permutation of (1234) and minus the unity for an odd permutation of (1234) and zero for a sequence that is not a permutation of (1234) at all. As this set of coefficients is completely antisymmetric with a number of indices that is equal to the dimension, we have that it has only one independent component, transforming as

$$\frac{\partial \boldsymbol{\alpha}^{i\_1}}{\partial \boldsymbol{\alpha}^{\prime i\_1'}} \frac{\partial \boldsymbol{\alpha}^{i\_2}}{\partial \boldsymbol{\alpha}^{\prime i\_2'}} \frac{\partial \boldsymbol{\alpha}^{i\_3}}{\partial \boldsymbol{\alpha}^{\prime i\_3'}} \frac{\partial \boldsymbol{\alpha}^{i\_4}}{\partial \boldsymbol{\alpha}^{\prime i\_4'}} \mathbf{e}\_{\boldsymbol{i}\_1 \boldsymbol{i}\_2 \boldsymbol{i}\_3 \boldsymbol{i}\_4} = \boldsymbol{e}\_{\boldsymbol{i}\_1' \boldsymbol{i}\_2' \boldsymbol{i}\_3' \boldsymbol{i}\_4'} \boldsymbol{\alpha}^{\prime}$$

for a given *α* function to be determined. In addition, because the determinant of any generic matrix can always be written in terms of these coefficients according to the expression given by det*M* = Σ*<sup>i</sup> j ǫi*1 *<sup>i</sup>*2*i*3*i*<sup>4</sup> *<sup>M</sup>*1*i*<sup>1</sup> *<sup>M</sup>*2*i*<sup>2</sup> *<sup>M</sup>*3*i*<sup>3</sup> *<sup>M</sup>*4*i*<sup>4</sup> , then

$$\det \frac{\partial \mathbf{x}}{\partial \mathbf{x}'} = \frac{\partial \mathbf{x}^{i\_1}}{\partial \mathbf{x}'^1} \frac{\partial \mathbf{x}^{i\_2}}{\partial \mathbf{x}'^2} \frac{\partial \mathbf{x}^{i\_3}}{\partial \mathbf{x}'^3} \frac{\partial \mathbf{x}^{i\_4}}{\partial \mathbf{x}'^4} \mathbf{e}\_{i\_1 i\_2 i\_3 i\_4} = \mathbf{e}\_{1234} \mathbf{a} = \mathbf{a}'$$

furnishing the *α* function. Thus, we have

$$\mathfrak{e}\_{i'\_1 i'\_2 i'\_3 i'\_4} = \det \frac{\partial \mathfrak{x'}}{\partial \mathfrak{x}} \frac{\partial \mathfrak{x'}^i}{\partial \mathfrak{x'}^{i'\_1}\_1} \frac{\partial \mathfrak{x'}^2}{\partial \mathfrak{x'}^{i'\_2}\_2} \frac{\partial \mathfrak{x'}^3}{\partial \mathfrak{x'}^{i'\_3}\_3} \frac{\partial \mathfrak{x'}^4}{\partial \mathfrak{x'}^{i'\_4}\_4} \mathfrak{e}\_{i\_1 i\_2 i\_3 i\_4}$$

which is non-tensorial, but its non-tensoriality perfectly matches that of the determinant of the metric. Therefore, we have that they compensate in the combined form

$$(g^{\frac{1}{2}}\epsilon\_{\alpha\nu\sigma\tau})' = \text{sign } \det \left| \frac{\partial x'}{\partial x} \middle| \frac{\partial x^{\beta}}{\partial x'^{\alpha}} \frac{\partial x^{\mu}}{\partial x'^{\nu}} \frac{\partial x^{\rho}}{\partial x'^{\sigma}} \frac{\partial x^{\rho}}{\partial x'^{\tau}} (g^{\frac{1}{2}}\epsilon\_{\beta\mu\theta\rho})\right|$$

which is in fact the transformation that defines a pseudo-tensorial field. Notice, however, that, if we were to define the tensor with all lower indices as

$$
\varepsilon\_{\mathfrak{A}\vee\sigma\tau} = \epsilon\_{\mathfrak{a}\vee\sigma\tau} |g|^{\frac{1}{2}},
$$

the correspondent tensor with all upper indices would be given according to the following expression:

$$
\epsilon^{\kappa\nu\sigma\tau} = \epsilon^{\kappa\nu\sigma\tau} |\g|^{-\frac{1}{2}}
$$

in order for it to be consistently defined. This difference is necessary, as it can be seen from the fact that the quantity

$$\begin{array}{c} \varepsilon^{i\_{1}i\_{2}i\_{3}i\_{4}} \varepsilon\_{j\_{1}j\_{2}j\_{3}j\_{4}} = -\mathsf{det}\begin{vmatrix} \delta^{i\_{1}}\_{j\_{1}} & \delta^{i\_{2}}\_{j\_{1}} & \delta^{i\_{3}}\_{j\_{1}} & \delta^{i\_{4}}\_{j\_{1}} \\ \delta^{i\_{1}}\_{j\_{2}} & \delta^{i\_{2}}\_{j\_{2}} & \delta^{i\_{3}}\_{j\_{2}} & \delta^{i\_{4}}\_{j\_{2}} \\ \delta^{i\_{1}}\_{j\_{3}} & \delta^{i\_{2}}\_{j\_{3}} & \delta^{i\_{3}}\_{j\_{3}} & \delta^{i\_{4}}\_{j\_{3}} \\ \delta^{i\_{1}}\_{j\_{4}} & \delta^{i\_{2}}\_{j\_{4}} & \delta^{i\_{3}}\_{j\_{4}} & \delta^{i\_{4}}\_{j\_{4}} \end{vmatrix} \end{array}$$

as it is very easy to check by performing a straightforward substitution and making all the direct calculations.

To summarize, the object *δ β <sup>α</sup>* that is unity or zero according to whether the value of its indices is equal or not is the *unity tensor* mentioned. We assume the existence of two tensors *gακ* and *g ακ* symmetric and such that

$$\mathcal{g}^{\sigma\mu}\mathcal{g}\_{\kappa\sigma} = \delta^{\mu}\_{\kappa} \tag{9}$$

 

called *metric tensors*. In addition, we define

$$
\delta\_{j\_0 j\_1 j\_2 j\_3}^{i\_0 i\_1 i\_2 i\_3} = \det \begin{vmatrix}
\delta\_{j\_0}^{i\_0} & \delta\_{j\_0}^{i\_1} & \delta\_{j\_0}^{i\_2} & \delta\_{j\_0}^{i\_3} \\
\delta\_{j\_1}^{i\_0} & \delta\_{j\_1}^{i\_1} & \delta\_{j\_1}^{i\_2} & \delta\_{j\_1}^{i\_3} \\
\delta\_{j\_2}^{i\_0} & \delta\_{j\_2}^{i\_1} & \delta\_{j\_2}^{i\_2} & \delta\_{j\_2}^{i\_3} \\
\delta\_{j\_3}^{i\_0} & \delta\_{j\_3}^{i\_1} & \delta\_{j\_3}^{i\_2} & \delta\_{j\_3}^{i\_3}
\end{vmatrix} \tag{10}
$$

as a *completely antisymmetric unity tensor*. The quantity given by *ǫi*0*i*<sup>1</sup> *i*2*i*3 equal to the unity, minus unity, or zero according to whether (*i*0*i*1*i*2*i*3) is an even, odd, or not a permutation of (0123) can be taken with the determinant of the metric det(*gµν*) =*g* to define

$$
\epsilon^{\text{a}\nu\sigma\tau} = \epsilon^{\text{a}\nu\sigma\tau} |\mathbf{g}|^{-\frac{1}{2}} \tag{11}
$$

as well as

$$
\varepsilon\_{\text{av}\sigma\tau} = \varepsilon\_{\text{av}\sigma\tau} |g|^{\frac{1}{2}} \tag{12}
$$

which are completely antisymmetric and such that

$$
\varepsilon^{i\_0 i\_1 i\_2 i\_3} \varepsilon\_{j\_0 j\_1 j\_2 j\_3} = -\delta^{i\_0 i\_1 i\_2 i\_3}\_{j\_0 j\_1 j\_2 j\_3} \tag{13}
$$

called *completely antisymmetric pseudo-tensors*. When a tensor with at least one index is multiplied by the metric tensor and the index is contracted with one index of the metric tensor, the result is a tensor in which the index has been *vertically moved*. In particular, if a tensor that is completely antisymmetric in *k* indices is multiplied by the completely antisymmetric pseudo-tensors and the *k* indices of the tensor are contracted with *k* indices of the completely antisymmetric pseudo-tensors, the result is a pseudo-tensor antisymmetric in (4 − *k*) indices called *dual*.

We are now at a point where we have defined for tensors a covariant operation that respects all rules of differentiation as well as the tensorial structure and an operation for the vertical re-configuration of tensorial indices, and we may wonder what happens when both operations are taken in parallel. More precisely, if the vertical index configuration cannot change the information content of a tensor, then this must be true for any tensor, and, in particular, if the tensor is the covariant derivative of some other tensor. Consequently, it must be possible to define

$$\mathcal{g}^{\alpha\beta} D\_{\mu} T^{\nu\ldots\zeta}\_{\beta\rho\sigma\ldots\theta} = D\_{\mu} T^{\alpha\nu\ldots\zeta}\_{\rho\sigma\ldots\theta}$$

which therefore implies

$$\begin{aligned} D\_{\mu}T^{\alpha\nu\ldots\tilde{\mathfrak{z}}}\_{\rho\sigma\ldots\theta} &= D\_{\mu}(g^{\alpha\beta}T^{\nu\ldots\tilde{\mathfrak{z}}}\_{\beta\rho\sigma\ldots\theta}) = D\_{\mu}g^{\alpha\beta}T^{\nu\ldots\tilde{\mathfrak{z}}}\_{\beta\rho\sigma\ldots\theta} + \\ &+ g^{\alpha\beta}D\_{\mu}T^{\nu\ldots\tilde{\mathfrak{z}}}\_{\beta\rho\sigma\ldots\theta} \end{aligned}$$

so that we are left with the equation

$$D\_{\mu} \mathcal{g}^{\alpha \beta} T^{\nu \ldots \zeta}\_{\beta \rho \sigma \ldots \theta} = 0$$

for any tensor, implying *Dµg αβ* =0 as well. This means that the metric tensor is covariantly constant. Conditions of vanishing of the covariant derivative of the metric tensor mean that the irrelevance of the indices disposition must be valid regardless of the differential order of the tensor. If we were to follow the common approach defining the metric first, these conditions would mean that the metric structure and the local structure will have to be independent. This is reasonable since, if a vector is constant, its norm should be constant too. It is interesting to notice that, since we have two types of covariant derivatives and because the present arguments hold, regardless of the specific covariant derivative, then we have to assume that both covariant derivatives of the metric tensor vanish as *Dµgαβ* = ∇*µgαβ* =0 in general. In particular, we have that *D<sup>θ</sup> εαβµν* = ∇*<sup>θ</sup> εαβµν* =0 hold as well. If we are insisting that this happen, then there are very remarkable consequences that follow. To see this, expand

$$0 = D\_{\rho} \mathbf{g}\_{\alpha \beta} = \partial\_{\rho} \mathbf{g}\_{\alpha \beta} - \mathbf{g}\_{\alpha \mu} \Gamma^{\mu}\_{\beta \rho} - \mathbf{g}\_{\mu \beta} \Gamma^{\mu}\_{\alpha \rho}$$

and take the three different indices permutations combined together with the definition of torsion to get

$$\begin{aligned} \Gamma^{\rho}\_{\alpha\beta} &= \frac{1}{2} \mathcal{Q}^{\rho}\_{\ \alpha\beta} + \frac{1}{2} (\mathcal{Q}\_{\alpha\beta}{}^{\rho} + \mathcal{Q}\_{\beta\alpha}{}^{\rho}) + \\ &+ \frac{1}{2} \mathcal{g}^{\rho\mu} \left( \partial\_{\beta} \mathcal{g}\_{\alpha\mu} + \partial\_{\alpha} \mathcal{g}\_{\mu\beta} - \partial\_{\mu} \mathcal{g}\_{\alpha\beta} \right) \end{aligned}$$

in which *Qρασ* is the torsion tensor antisymmetric in the two lower indices, while (*Qαβρ*+ *Qβαρ*) is a tensor symmetric in those indices, whereas the remaining coefficients written in terms of the partial derivatives of the metric tensor transform as a connection and they are symmetric in those very indices. This expression shows that the most general connection can be decomposed in terms of the torsion plus a symmetric connection, as we already knew from expression (3), but, in addition, it tells us the explicit form of Λ *ρ αβ* as given by a symmetric combination of two torsions plus a symmetric connection entirely written in terms of the metric. It is essential to note that, if we want all possible connections to give rise to covariant derivatives, which, once applied onto the metric, give zero, then we have to restrict the torsion to verify

$$Q\_{a\beta\rho} = -Q\_{\beta\alpha\rho}$$

spelling its complete antisymmetry [10]. The condition of metric-compatible connection extended to all connections implies the torsion to be completely antisymmetric, once again establishing a link between two structures that are a priori unrelated. The complete antisymmetry of torsion is equivalent to the existence of a single symmetric part of the connection, and therefore to the existence of a unique connection writable in terms of the metric alone. It is a remarkable fact that the torsion tensor could be reduced to be completely antisymmetric by employing a number of unrelated arguments as those presented in [11–14] and, although torsion might well not display such a symmetry, it is certainly intriguing to argue what the consequences are of this condition. We will see that some of these consequence are of paramount importance next.

Thus, we summarize by saying that the torsion tensor with all lower indices is taken to be completely antisymmetric and therefore it is possible to write it according to

$$Q\_{a\sigma\nu} = \frac{1}{6} \mathcal{W}^{\mu} \varepsilon\_{\mu\alpha\sigma\nu} \tag{14}$$

in terms of the *W<sup>µ</sup>* pseudo-vector, therefore called the torsion pseudo-vector, while the connection

$$
\Lambda^{\rho}\_{a\beta} = \frac{1}{2} g^{\rho\mu} \left( \partial\_{\beta} \gspace{0.} \partial\_{\beta} \gspace{0.} g\_{a\mu} + \partial\_{a} \gspace{0.} g\_{\mu\beta} - \partial\_{\mu} \gspace{0.} g\_{a\beta} \right) \tag{15}
$$

is symmetric and written entirely in terms of the partial derivatives of the metric tensor and, for this reason called the *metric connection*, so that

$$\Gamma^{\rho}\_{\alpha\beta} = \frac{1}{2} \mathcal{g}^{\rho\mu} \left[ (\partial\_{\beta} \mathcal{g}\_{\alpha\mu} + \partial\_{\alpha} \mathcal{g}\_{\mu\beta} - \partial\_{\mu} \mathcal{g}\_{\alpha\beta}) + \frac{1}{6} \mathcal{W}^{\nu} \varepsilon\_{\nu\mu\alpha\beta} \right] \tag{16}$$

is the most general connection and such a decomposition is equivalent to the validity of the following conditions:

$$\nabla\_{\theta} \varepsilon\_{\alpha \beta \mu \upsilon} \equiv \mathcal{D}\_{\theta} \varepsilon\_{\alpha \beta \mu \upsilon} = 0 \tag{17}$$

$$\nabla\_{\mu} \mathcal{g}\_{a\beta} \equiv D\_{\mu} \mathcal{g}\_{a\beta} = 0 \tag{18}$$

called *metric-compatibility conditions for the connection*.

Thus far, we have defined tensors and the properties compatible with the derivation. It is now the time to see what happens when we go to a following order derivative.

We may proceed to calculate the commutator of two derivatives, which in the particular case of vectors is

$$\begin{aligned} [D\_{\alpha}, D\_{\beta}]T^{\sigma} &= (\Gamma^{\rho}\_{\kappa\beta} - \Gamma^{\rho}\_{\beta\alpha})D\_{\rho}T^{\sigma} + \\ + (\partial\_{\alpha}\Gamma^{\sigma}\_{\kappa\beta} - \partial\_{\beta}\Gamma^{\sigma}\_{\kappa\alpha} + \Gamma^{\rho}\_{\kappa\beta}\Gamma^{\sigma}\_{\rho\alpha} - \Gamma^{\rho}\_{\kappa\alpha}\Gamma^{\sigma}\_{\rho\beta})T^{\kappa} \end{aligned}$$

with no second derivatives. The only derivative term left is proportional to the torsion tensor *Q ρ µα* plus another

$$G^{\sigma}\_{\kappa\alpha\beta} = \partial\_{\mathfrak{A}}\Gamma^{\sigma}\_{\kappa\beta} - \partial\_{\beta}\Gamma^{\sigma}\_{\kappa\alpha} + \Gamma^{\rho}\_{\kappa\beta}\Gamma^{\sigma}\_{\rho\alpha} - \Gamma^{\rho}\_{\kappa\alpha}\Gamma^{\sigma}\_{\rho\beta}$$

which, although written in terms of the connection alone, is a tensor. With these expressions, we have

$$[D\_{\alpha}, D\_{\beta}]T^{\sigma} = Q^{\rho}\_{\alpha\beta} D\_{\rho}T^{\sigma} + G^{\sigma}\_{\kappa\alpha\beta}T^{\kappa}$$

giving the commutator of vectors in particular. As it has been done for the connection and the most general covariant derivative, the interesting thing is that the definition of tensor *G σ καβ* can be used in the most general case of the commutator of covariant derivatives. We also have

$$\begin{aligned} (\partial \partial T)\_{\alpha \beta \rho \dots \mu} &= \partial\_{[\alpha} (\partial T)\_{\beta \rho \dots \mu]} = \partial\_{[\alpha} \partial\_{[\beta} T\_{\rho \dots \mu]]} = 0 \\ &= \partial\_{[\alpha} \partial\_{\beta} T\_{\rho \dots \mu]} = 0 \end{aligned}$$

because partial derivatives always commute and therefore their commutator is always zero.Before we have had the opportunity to briefly talk about external calculus, where the external derivatives are used to calculate the border of a manifold, and the above expression refers to the fact that the border has a border that vanishes, or that there is no border of a border. Once again, apart from curiosity, there is no need to deepen these concepts in the following.

To summarize, from the connection, we may calculate

$$G^{\sigma}\_{\ \kappa\alpha\beta} = \partial\_{\mathfrak{a}} \Gamma^{\sigma}\_{\kappa\beta} - \partial\_{\beta} \Gamma^{\sigma}\_{\kappa\alpha} + \Gamma^{\sigma}\_{\rho\alpha} \Gamma^{\rho}\_{\kappa\beta} - \Gamma^{\sigma}\_{\rho\beta} \Gamma^{\rho}\_{\kappa\alpha} \tag{19}$$

which is a tensor antisymmetric in the last two indices and verifying the following cyclic permutation condition

$$\begin{aligned} D\_{\kappa}Q^{\rho}\_{\ \mu\nu} + D\_{\nu}Q^{\rho}\_{\ \kappa\mu} + D\_{\mu}Q^{\rho}\_{\ \nu\kappa} + \\ + Q^{\pi}\_{\ \nu\kappa}Q^{\rho}\_{\ \mu\pi} + Q^{\pi}\_{\ \mu\nu}Q^{\rho}\_{\ \kappa\pi} + Q^{\pi}\_{\ \kappa\mu}Q^{\rho}\_{\ \nu\pi} - \\ -G^{\rho}\_{\ \kappa\nu\mu} - G^{\rho}\_{\ \mu\kappa\nu} - G^{\rho}\_{\ \nu\mu\kappa} \equiv 0 \end{aligned} \tag{20}$$

called the *curvature tensor* and decomposable as

$$\begin{split} \mathbf{G}^{\sigma}\_{\ \kappa\alpha\beta} &= \mathbf{R}^{\sigma}{}\_{\ \kappa\alpha\beta} + \frac{1}{2} (\nabla\_{\alpha} \mathbf{Q}^{\sigma}{}\_{\ \kappa\beta} - \nabla\_{\beta} \mathbf{Q}^{\sigma}{}\_{\ \kappa\alpha}) + \\ &+ \frac{1}{4} (\mathbf{Q}^{\sigma}{}\_{\ \rho\alpha} \mathbf{Q}^{\rho}{}\_{\ \kappa\beta} - \mathbf{Q}^{\sigma}{}\_{\ \rho\beta} \mathbf{Q}^{\rho}{}\_{\ \kappa\alpha}) \end{split} \tag{21}$$

in terms of torsion and

$$R^{\sigma}\_{\ \kappa\alpha\beta} = \partial\_{\mathfrak{u}} \Lambda^{\sigma}\_{\kappa\beta} - \partial\_{\beta} \Lambda^{\sigma}\_{\kappa\mathfrak{u}} + \Lambda^{\sigma}\_{\rho\mathfrak{u}} \Lambda^{\rho}\_{\kappa\mathfrak{f}} - \Lambda^{\sigma}\_{\rho\mathfrak{f}} \Lambda^{\rho}\_{\kappa\mathfrak{u}} \tag{22}$$

as a tensor antisymmetric in the last two indices and such that it verifies the cyclic permutation condition

$$\left(\boldsymbol{R}^{\rho}\_{\;\;\nu\;\mu} + \boldsymbol{R}^{\rho}\_{\;\;\mu\&\nu} + \boldsymbol{R}^{\rho}\_{\;\;\nu\;\mu\&} \equiv 0\tag{23}$$

called *metric curvature tensor*. By employing torsion and curvature, it is possible to demonstrate that we have

$$\begin{aligned} \left[D\_{\mu\nu}D\_{\nu}\right]T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}}&=\mathcal{Q}^{\eta}\_{\ \mu\nu}D\_{\eta}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}}+\\ &+\sum\_{k=1}^{k=j}\mathcal{G}^{\alpha\_{k}}\_{\ \sigma\mu\nu}T^{\alpha\_{1}\ldots\sigma\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\beta\_{i}}-\\ &-\sum\_{k=1}^{k=i}\mathcal{G}^{\sigma}\_{\ \beta\_{k}\mu\nu}T^{\alpha\_{1}\ldots\alpha\_{j}}\_{\beta\_{1}\ldots\sigma\ldots\beta\_{i}}\end{aligned} \tag{24}$$

as the expression for *commutator of covariant derivatives* of the tensor field. In particular, we have that

$$
\partial \partial T = 0 \tag{25}
$$

which is valid in the most general circumstance.

We have the validity of the following decomposition

$$\begin{split} R\_{\mathsf{k}\rho\boldsymbol{a}\mu} &= \frac{1}{2} (\partial\_{\mathsf{a}}\partial\_{\rho}g\_{\mu\kappa} - \partial\_{\boldsymbol{\mu}}\partial\_{\rho}g\_{\kappa\alpha} + \partial\_{\boldsymbol{\mu}}\partial\_{\kappa}g\_{\kappa\rho} - \partial\_{\boldsymbol{\kappa}}\partial\_{\boldsymbol{a}}g\_{\kappa\rho}) + \\ &+ \frac{1}{4} \mathsf{g}^{\sigma\boldsymbol{\nu}} [(\partial\_{\rho}g\_{\mu\nu} + \partial\_{\boldsymbol{a}}g\_{\rho\nu} - \partial\_{\boldsymbol{\nu}}g\_{\rho\alpha})(\partial\_{\boldsymbol{\kappa}}g\_{\mu\sigma} + \partial\_{\boldsymbol{\mu}}g\_{\kappa\sigma} - \partial\_{\boldsymbol{\sigma}}g\_{\kappa\mu}) - \\ &- (\partial\_{\boldsymbol{\rho}}g\_{\mu\nu} + \partial\_{\boldsymbol{\mu}}g\_{\rho\nu} - \partial\_{\boldsymbol{\nu}}g\_{\rho\mu}) \left(\partial\_{\boldsymbol{\kappa}}g\_{\mu\sigma} + \partial\_{\boldsymbol{a}}g\_{\kappa\sigma} - \partial\_{\boldsymbol{\sigma}}g\_{\kappa\mu}\right)] \end{split} \tag{26}$$

showing the antisymmetry also in the first two indices as well as the symmetry involving all four indices

$$R\_{\rho\kappa\mu\nu} = R\_{\mu\nu\rho\kappa} \tag{27}$$

and, as a consequence, the metric curvature tensor has one independent contraction *Rµσ* =*R ρ µρσ*, which is symmetric and called *Ricci metric curvature tensor* with contraction *R*=*Rµσg µσ* called *Ricci metric curvature scalar*, so that, with torsion, we can write

*Gκραµ* = <sup>1</sup> 2 (*∂α∂ρgµκ*−*∂µ∂ρgκα*+*∂µ∂κgαρ*−*∂κ∂αgµρ*) + +<sup>1</sup> 4 *g σν*[(*∂ρgαν*+*∂αgρν*−*∂νgρα*)(*∂κgµσ*+*∂µgκσ*−*∂σgκµ*) <sup>−</sup> −(*∂ρgµν*+*∂µgρν*−*∂νgρµ*)(*∂κgασ*+*∂αgκσ*−*∂σgκα*)] + (28) + <sup>1</sup> 12∇*ηW<sup>σ</sup>* (*gαηεσκρµ*−*gµηεσκρα*)+ <sup>1</sup> <sup>144</sup> [*WσW<sup>σ</sup>* (*gµρgακ*−*gµκgαρ*) + +(*WαWρgµκ*−*WµWρgακ*+*WµWκgαρ*−*WαWκgµρ*)]

showing the antisymmetry in the first two indices, and, as a consequence, it has one independent contraction chosen as *Gµσ* =*G ρ µρσ* called *Ricci curvature tensor* whose contraction *G*=*Gµσg µσ* is called *Ricci curvature scalar*.

In addition, finally, we may consider the cyclic permutation of commutator of commutators of covariant derivatives and see that the results are geometric identities.

In general, we have that we can write

$$D\_{\mu}G^{\nu}{}\_{\mu\xi\rho} + D\_{\kappa}G^{\nu}{}\_{\mu\rho} + D\_{\rho}G^{\nu}{}\_{\mu\kappa} + \\
$$

$$+ G^{\nu}{}\_{\mu\beta\mu}Q^{\beta}{}\_{\rho\kappa} + G^{\nu}{}\_{\mu\beta\kappa}Q^{\beta}{}\_{\mu\rho} + G^{\nu}{}\_{\mu\beta\rho}Q^{\beta}{}\_{\kappa\mu} \equiv 0\tag{29}$$

for torsion and curvature valid as a geometric identity.

Thus far, we have introduced the concept of tensor and the way to move its indices, which we recall were coordinate indices. Coordinate indices are important since they are the type of indices involved in differentiation. However, on the other hand, tensors in coordinate indices always feel the specificity of the coordinate system. Tensorial equations do remain formally the same in all coordinate system, but the tensors themselves change in content while changing the coordinate system. The only types of tensors which, also in content, remain the same in all of the coordinate systems are the tensors that are identically equal to zero and the scalars. Zero tensors offer little information, but scalars can be

used to build a formalism in which tensors can be rendered, both in form and in content, completely invariant. This formalism is known as Lorentz formalism.

In Lorentz formalism, the idea is that of introducing a basis of vectors *ξ α <sup>a</sup>* having two types of indices: one type of indices (Greek) is the usual coordinate index referring to the component of the vector, whereas the other type of indices (Latin) is a new Lorentz index referring to which vector of the basis we are considering. Under the point of view of coordinate transformations, the coordinate index ensures the transformation law of a vector, but clearly the other index ensures some different type of transformation that we will next find to be a Lorentz transformation.

Consider, for example, the tensor given by *Tασ* and multiply it by two of the vectors *ξ α <sup>a</sup>* of the basis contracting the coordinate indices together: so *Tασξ α a ξ σ <sup>s</sup>* =*Tas* is an object that according to a coordinate transformation law does not transform at all, thus it is completely invariant, and this is exactly what we wanted. For one tensor with upper indices, the procedure would be the same but just made in terms of the covectors *ξ a <sup>α</sup>* as clear. Converting a coordinate index to a Lorentz index and then back to the coordinate index requires that *ξ α b ξ c <sup>α</sup>* = *δ c b* and *ξ α k ξ k <sup>σ</sup>* = *δ α <sup>σ</sup>* as a simple consistency condition. Finally, the operation for moving Lorentz indices is performed in terms of the metric tensor in Lorentz form *gασξ α a ξ σ <sup>s</sup>* =*gas*, but, because we can always ortho-normalize the basis, the metric tensor in Lorentz form is just the Minkowskian matrix *gas* =*ηas* as it is well known indeed. Once the basis *ξ σ a* is assigned, we may pass to another basis *ξ* ′*σ a* linked to the initial according to the transformation *ξ* ′*σ <sup>a</sup>* =Λ*<sup>b</sup> a ξ σ <sup>b</sup>* with <sup>Λ</sup>*<sup>b</sup> a* chosen as to preserve the structure of the Minkowskian matrix and so such that it has to yield *η* = Λ*η*Λ*<sup>T</sup>* known as Lorentz transformation and justifying the name of the formalism.

In conclusion, after that, the coordinate tensors are converted into the Lorentz tensors, they are scalars under a general coordinate transformation but tensors under the Lorentz transformations. In doing so, we have converted the most general formalism into an equivalent formalism in which, however, the structure of the transformation now is very specific, and it can be made explicit. It is, in fact, known from the theory of Lie groups that any continuous transformation is writable according to

$$
\Lambda = e^{\frac{1}{2}\sigma^{ab}\theta\_{ab}}
$$

in which *θab* = −*θba* are the parameters while *σab* = −*σba* are the generators and which verify specific commutation relationships that depend on the specific transformation alone. In the case of Lorentz transformation, it is known that we have six parameters and six generators given by

$$(\sigma\_{ab})^i\_j = \delta^i\_a \eta\_{jb} - \delta^i\_b \eta\_{ja}$$

and verifying

$$[\sigma\_{ab}, \sigma\_{cd}] = \eta\_{ad}\sigma\_{bc} - \eta\_{ac}\sigma\_{bd} + \eta\_{bc}\sigma\_{ad} - \eta\_{bd}\sigma\_{ac}$$

in general. While the generators are peculiar of this so-called real representation, the commutations relationship are meant to be a general character of the Lorentz transformation. As such, they will always be the same for any representation of Lorentz transformations. This shall be the Lorentz transformation that we will employ next.

We may condense everything into the following statements, starting from the fact that given a Lorentz transformation Λ the set of functions *T a*1 ...*a<sup>i</sup> r*1 ...*r<sup>j</sup>* transforming as

$$T\_{r\_1'\dots r\_n'}^{a\_1'\dots a\_m'} = (\Lambda^{-1})\_{r\_1'}^{r\_1} \dots (\Lambda^{-1})\_{r\_n'}^{r\_n} (\Lambda)\_{a\_1}^{a\_1'} \dots (\Lambda)\_{a\_m}^{a\_m'} T\_{r\_1\dots r\_n}^{a\_1\dots a\_m} \tag{30}$$

is a *tensor in Lorentz formalism*. Compared to the coordinate formalism, symmetry properties and contractions, as well as all algebraic operations, are given analogously.

However, again, Lorentz transformations can be local and so differential operations must be defined by introducing a connection. As we have done before, the connection must be introduced in general in terms of its transformation.

Therefore, once again, we summarize by saying that the set of functions Ω*<sup>a</sup> bµ* such that, under a general coordinate transformation, transforming as a lower Greek index vector and under a Lorentz transformation transforming as

$$
\Omega\_{b'\nu}^{\prime a'} = \Lambda\_a^{a'} \left[ \Omega\_{b\nu}^a - (\Lambda^{-1})\_k^a (\partial\_\nu \Lambda)\_b^k \right] (\Lambda^{-1})\_{b'}^b \tag{31}
$$

is called *spin connection*, and no decomposition nor in particular any torsion can be defined as no transposition of indices of different types is defined. With it, we have

$$\begin{split} D\_{\mu}T\_{r\_{1}\ldots r\_{j}}^{a\_{1}\ldots a\_{i}} &= \partial\_{\mu}T\_{r\_{1}\ldots r\_{j}}^{a\_{1}\ldots a\_{i}} + \sum\_{k=1}^{k=i} \Omega\_{p\mu}^{a\_{k}} T\_{r\_{1}\ldots r\_{j}}^{a\_{1}\ldots p\ldots a\_{i}} - \\ &- \sum\_{k=1}^{k=j} \Omega\_{r\_{k}\mu}^{p} T\_{r\_{1}\ldots p\ldots r\_{j}}^{a\_{1}\ldots a\_{i}} \end{split} \tag{32}$$

as *covariant derivative* of tensors in Lorentz formalism.

As we have anticipated, the passage to this formalism is done with the *ξ a <sup>σ</sup>* and *ξ σ a* vectors while the vertical movement of Latin indices is done with the *η ab* matrix.

Thus, the passage from general coordinate formalism to the Lorentz formalism is made via the introduction of the bases of vectors *ξ a <sup>σ</sup>* and *ξ σ <sup>a</sup>* dual of one another

$$
\mathfrak{F}^{a}\_{\mu} \mathfrak{F}^{\mu}\_{r} = \delta^{a}\_{r} \tag{33}
$$

$$
\mathfrak{F}^{a}\_{\mu} \mathfrak{F}^{\rho}\_{a} = \delta^{\rho}\_{\mu} \tag{34}
$$

called *tetrad fields* and such that they verify the pair of ortho-normality conditions given by

$$\mathcal{g}^{a\sigma} \mathfrak{f}^a\_a \mathfrak{f}^b\_{\sigma} = \eta^{ab} \tag{35}$$

$$g\_{a\sigma} \mathfrak{f}\_a^{\alpha} \mathfrak{f}\_b^{\sigma} = \eta\_{ab} \tag{36}$$

as *η* are the *Minkowskian matrices*, preserved by Lorentz transformations. With the dual bases, ortho-normal with respect to the Minkowskian matrices, we can take a tensor in coordinate formalism with at least one Greek index and multiply it by the basis contracting one Greek index with the Greek index of the bases therefore obtaining the tensor in Lorentz formalism with a Latin index, and with a vertical movement of Latin indices which is performed in terms of the Minkowskian matrix as it is expected.

Notice that, if these two formalisms are perfectly equivalent, then their covariant derivatives should be equivalent and in particular we should be able from the most general connection to derive the spin connection. Upon requiring that *Dµξ α <sup>a</sup>* =0 as well as *Dµηab* =0, we have exactly this.

In fact, in terms of the most general coordinate connection and tetrad fields, we can always write

$$
\Omega^{a}\_{b\mu} = \mathfrak{J}^{\nu}\_{b} \mathfrak{J}^{a}\_{\rho} \left( \Gamma^{\rho}\_{\nu\mu} - \mathfrak{J}^{\rho}\_{k} \partial\_{\mu} \mathfrak{J}^{\kappa}\_{\nu} \right) \tag{37}
$$

antisymmetric in the Lorentz indices and such that, from it, we can derive the torsion tensor according to

$$Q^{a}\_{\ \mu\nu} = - (\partial\_{\mu} \mathfrak{J}^{a}\_{\nu} - \partial\_{\nu} \mathfrak{J}^{a}\_{\mu} + \mathfrak{J}^{b}\_{\nu} \Omega^{a}\_{b\mu} - \mathfrak{J}^{b}\_{\mu} \Omega^{a}\_{b\nu}) \tag{38}$$

as it is easy to see, and we have that conditions (37) and <sup>Ω</sup>*ab<sup>µ</sup>* = −Ω*ba<sup>µ</sup>* are respectively equivalent to

$$D\_{\mu}\xi\_{\mu}^{r} = 0\tag{39}$$

$$D\_{\mu}\eta\_{ab} = 0\tag{40}$$

as general *coordinate-Lorentz compatibility conditions*.

In Lorentz formalism, from the spin connection, we get

$$G^{a}\_{\
ba\beta} = \partial\_{a}\Omega^{a}\_{b\beta} - \partial\_{\beta}\Omega^{a}\_{ba} + \Omega^{a}\_{ka}\Omega^{k}\_{b\beta} - \Omega^{a}\_{k\beta}\Omega^{k}\_{ba} \tag{41}$$

as the *curvature tensor*. Then, we have that

$$\begin{aligned} [D\_{\mu\nu}D\_{\nu}]T^{r\_1\dots r\_j} &= \mathcal{Q}^{\eta}{}\_{\mu\nu}D\_{\eta}T^{r\_1\dots r\_j} + \\ &+ \sum\_{k=1}^{k=j} \mathcal{G}^{r\_k}{}\_{p\mu\nu}T^{r\_1\dots p\dots r\_j} \end{aligned} \tag{42}$$

is the general coordinate covariant *commutator of covariant derivatives* of the tensor field in Lorentz formalism.

As it should be expected by now, we have that

$$\mathbf{G}^{a}\_{\ \flat \mu \nu} = \mathfrak{E}^{a}\_{a} \mathfrak{E}^{\beta}\_{b} \mathbf{G}^{a}\_{\ \flat \mu \nu} \tag{43}$$

showing that the curvature tensor in Lorentz formalism is antisymmetric both in coordinate indices and in Lorentz indices, and so as a consequence the curvature also in this formalism has the same independent contractions which are therefore *Gb<sup>σ</sup>* = *G a <sup>b</sup>ρσξ ρ a* for the *Ricci curvature tensor* and *G*=*Gaσξ σ pη ap* for the *Ricci curvature scalar*.

After index renaming, we get

$$\begin{aligned} &D\_{\mu}G^{a}{}\_{j\kappa\rho} + D\_{\kappa}G^{a}{}\_{j\rho\mu} + D\_{\rho}G^{a}{}\_{j\mu\kappa} + \\ &+ G^{a}{}\_{j\beta\mu}Q^{\beta}{}\_{\rho\kappa} + G^{a}{}\_{j\beta\kappa}Q^{\beta}{}\_{\mu\rho} + G^{a}{}\_{j\beta\rho}Q^{\beta}{}\_{\kappa\mu} \equiv 0 \end{aligned} \tag{44}$$

with curvature in Lorentz form as a geometric identity.

In this way, we conclude the introduction of the most general covariant formalism with the further conversion into the specific Lorentz formalism, in which the Lorentz transformation has been made explicit in terms of its real representation. We will soon see that another representation is possible. However, before this, we introduce gauge fields.

Our main goal is going to be focusing on the fact that fields may be complex, and so it makes sense to ask what symmetries can be established for these fields: if a field is complex, there arises the issue of phase transformations and, correspondingly, it is possible to construct a calculus that is in all aspects analogous to the one we just built.

Thus, given a real function *α*, we have that a complex field that transforms according to the transformation

$$
\phi' = e^{iqa} \phi \tag{45}
$$

is called *gauge field* of *q charge*, with algebraic operations defined as for geometric tensors.

Let it be given a covector field *A<sup>ν</sup>* such that, for a phase transformation, it transforms according to the law

$$A'\_{\nu} = A\_{\nu} - \partial\_{\nu} \mathfrak{a} \tag{46}$$

then this vector is called *gauge potential*. With it,

$$D\_{\mu}\phi = \partial\_{\mu}\phi + iqA\_{\mu}\phi\tag{47}$$

is said to be the *gauge derivative* of the gauge field.

For the gauge fields, we may introduce the operation of complex conjugation without the necessity of introducing any additional structure, and hence, for a gauge field of *q* charge, the complex conjugate gauge field has −*q* charge.

There is no decomposition of the gauge potential into more fundamental elements. In fact, complex conjugation is compatible with gauge derivatives automatically.

From the gauge connection, we define

$$F\_{\mathfrak{a}\mathfrak{F}} = \partial\_{\mathfrak{a}} A\_{\mathfrak{F}} - \partial\_{\mathfrak{F}} A\_{\mathfrak{a}} \tag{48}$$

that is such that *F* = *∂A* and so it is a tensor which is antisymmetric and invariant by a gauge transformation called *gauge strength*. With it, we have that

$$[D\_{\mu\nu}D\_{\nu}]\phi = iqF\_{\mu\nu}\phi\tag{49}$$

is the *commutator of gauge derivatives* of gauge fields.

Clearly, the gauge strength cannot be decomposed in terms of more fundamental underlying structures.

Furthermore, we have that

$$
\partial\_\nu F\_{\nu\sigma} + \partial\_\sigma F\_{\nu\alpha} + \partial\_\mu F\_{\sigma\nu} = 0 \tag{50}
$$

or equivalently *∂F* = 0 as a gauge geometric identity.

There is a point that needs to be elucidated regarding the definition of the Maxwell strength. As this expression can be generalized up to

$$F\_{\alpha\beta} = \nabla\_{\alpha}A\_{\beta} - \nabla\_{\beta}A\_{\alpha\nu}$$

then one may wonder if some non-minimal coupling could be invoked to write it as

$$F'\_{\alpha\beta} = D\_{\alpha}A\_{\beta} - D\_{\beta}A\_{\alpha} = \nabla\_{\alpha}A\_{\beta} - \nabla\_{\beta}A\_{\alpha} + Q\_{\alpha\beta\rho}A^{\rho}$$

which would violate gauge invariance. Therefore, which one between *Fαβ* and *F* ′ *αβ* should be considered? The answer is actually quite simple conceptually, and it is that, in a theory of electrodynamics established within a purely geometric context, the Maxwell strength is not just a curl of a vector but the specific curl of a vector that comes as the formal expression of the curvature of two covariant derivatives. In this sense, it is clear that *F* ′ *αβ* as compared to *Fαβ* has a lesser geometric meaning. Moreover, the form *Fαβ* is also the one for which the geometric identities (50) called Cauchy identities are valid. In addition, so this is the only form that will interest us in the following.

This concludes the introduction of gauge fields, based on a parallel with geometric tensor. We shall now move to a following part in which these two formalisms will be merged into a single one known as spinorial formalism.

### 2.1.2. Spinorial Fields

In the previous parts, we have introduced tensor fields and the way to pass from coordinate into Lorentz indices, specifying that, with such a conversion, we also had the conversion of the most general coordinate transformation into the specific Lorentz transformation: the advantage of this specific Lorentz transformation is that, although it had been introduced in real representation, nevertheless, it can also be written in other representations like most notably the complex representation. In such representation, we will see that gauge fields find place naturally.

In order to find a Lorentz transformation in complex representation, we specify that these transformations are classified by semi-integer labels known as spin, and here we consider the simplest <sup>1</sup> 2 -spin case: so, for the complex generators, we select those whose irreducible form is given in terms of two-dimensional matrices. General results from the theory of Lie groups tell us that the Lorentz transformation can be written according to the following form:

$$\Lambda = e^{\frac{1}{2}\sigma^{ab}\theta\_{ab}}$$

where *θ ab* <sup>=</sup> <sup>−</sup>*<sup>θ</sup> ba* are the parameters as given above and *<sup>σ</sup>ab* <sup>=</sup> <sup>−</sup>*σba* are the generators verifying

$$[\sigma\_{ab\prime}\sigma\_{cd}] = \eta\_{ad}\sigma\_{bc} - \eta\_{ac}\sigma\_{bd} + \eta\_{bc}\sigma\_{ad} - \eta\_{bd}\sigma\_{ac}$$

as commutation relationships. The actual form for these Lorentz generators in the case of the complex irreducible two-dimensional matrices is known to be given in terms of the Pauli matrices

$$
\sigma^1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \sigma^2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \quad \sigma^3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
$$

according to

$$\begin{aligned} \sigma^{0A}\_{\pm} &= \pm \frac{1}{2} \sigma^A \\ \sigma\_{AB} &= -\frac{i}{2} \varepsilon\_{ABC} \sigma^C \end{aligned}$$

as a straightforward check would demonstrate. We notice that, in the passage from real to complex representation, a two-fold multiplicity has arisen since two opposite expressions are possible for the boosts and thus for the Lorentz transformation in full. This ambiguity can be overcome by having these two irreducible two-dimensional generators merged into a single reducible four-dimensional generators

$$
\begin{aligned}
\sigma^{0A} &= \frac{1}{2} \begin{pmatrix} -\sigma^A & 0 \\ 0 & \sigma^A \end{pmatrix} \\
\sigma\_{AB} &= -\frac{i}{2} \varepsilon\_{ABC} \begin{pmatrix} \sigma^\complement & 0 \\ 0 & \sigma^\complement \end{pmatrix}
\end{aligned}
$$

which still verify the Lorentz commutation algebra. Such a merging also has the advantage that, with four-dimensional matrices, it is possible to introduce

$$
\begin{pmatrix} \mathbb{I} & \mathbb{0} \\ \mathbb{I} & \mathbb{0} \\ \mathbb{0} & \mathbb{0} \end{pmatrix} = \gamma^0
$$
 
$$
\begin{pmatrix} 0 & \sigma^K \\ -\sigma^K & 0 \end{pmatrix} = \gamma^K
$$

in terms of which the four-dimensional generators are

$$\sigma^{ab} = \frac{1}{4} \left[ \gamma^a \gamma^b \right]\_{,}$$

and where

$$\{\gamma^a \gamma^b\} = 2\eta^{ab}\Box$$

in terms of the Minkowskian matrix. This way of writing four-dimensional matrices constitutes an advantage because we can see a manifest (1 + 3)-dimensional space–time form in the last two expressions. Then, it is possible to employ these last two expressions to derive a whole list of useful identities involving these matrices. To begin, we have

$$
\sigma\_{ab} = -\frac{i}{2} \varepsilon\_{abcd} \pi \sigma^{cd}
$$

which implicitly defines the *π* matrix. This matrix is the one usually indicated like a gamma with an index five as originally it was used to study five-dimensional theories, but, because we will always be in the space–time, the index five for us has no meaning and

so we will use a notation with no index at all. Notice that, with this definition, we have extinguished all possible matrices since the matrices

$$\mathbb{I} \qquad \gamma^a \qquad \sigma^{ab} \qquad \gamma^a \pi \qquad \pi$$

are 16 linearly independent matrices spanning the space of four-dimensional matrices, and so they form a basis for such a space. These matrices are called Clifford matrices and they will have great importance. We have that

$$\begin{aligned} \gamma\_0 \gamma\_a^\dagger \gamma\_0 &= \gamma\_a \\ \gamma\_0 \sigma\_{ab}^\dagger \gamma\_0 &= -\sigma\_{ab} \\ \pi^\dagger &= \pi \end{aligned}$$

specifying the behavior of the Clifford matrices under conjugation. By direct inspection, one can easily see that

$$
\gamma\_a \gamma\_b = \eta\_{ab} \mathbb{I} + 2\sigma\_{ab}
$$

as well as

$$
\gamma\_i \gamma\_j \gamma\_k = \gamma\_i \eta\_{jk} - \gamma\_j \eta\_{ik} + \gamma\_k \eta\_{ij} + i \varepsilon\_{ijkq} \pi \gamma^q
$$

showing that products of, however, many gamma matrices can always be reduced to the product of at most two of them. Therefore, there is no need to compute the product of three or more gamma matrices. Because *ε*<sup>0123</sup> =1, we have *π* =*iγ* 0*γ* 1*γ* 2*γ* <sup>3</sup> and so

$$\begin{aligned} \{\pi, \gamma\_a\} &= 0\\ [\pi, \sigma\_{ab}] &= 0 \end{aligned}$$

as expected. In fact, this representation is reducible, and then Schur's lemma ensures us that there must exist one matrix different from the identity commuting with all the generators of the group. By working with all the previous identities, one can find

$$\begin{aligned} [\gamma\_{i\prime}\sigma\_{jk}] &= \gamma\_k \eta\_{ij} - \gamma\_j \eta\_{ik} \\ \{\gamma\_{i\prime}\sigma\_{jk}\} &= i\varepsilon\_{ijkq}\pi\gamma\gamma^q \end{aligned}$$

and similarly

$$\begin{aligned} \{\sigma\_{ab\prime}\sigma\_{cd}\} &= \frac{1}{2}(\eta\_{ad}\eta\_{bc}\mathbb{I} - \eta\_{ac}\eta\_{bd}\mathbb{I} + i\varepsilon\_{abcd}\pi) \\ \{\sigma\_{ab\prime}\sigma\_{cd}\} &= \eta\_{ad}\sigma\_{bc} - \eta\_{ac}\sigma\_{bd} + \eta\_{bc}\sigma\_{ad} - \eta\_{bd}\sigma\_{ac} \end{aligned}$$

as other fundamental identities. The list may go on, but, for our purposes, there is no need to reach products with more gamma matrices. The last identity tells us that the *σab* matrices are the generators of the Lorentz algebra as expected. As already said, the parameters are the same we had in the Lorentz formalism since real and complex representations are merely two different forms of the same transformation. This transformation is thus given by

$$
\Lambda = e^{\frac{1}{2}\sigma^{ab}\theta\_{ab}}
$$

in its most general form. However, in view of studying complex fields, we know that the complex phase transformation *e iq<sup>α</sup>* must also be introduced. Therefore, we have

$$\Lambda e^{iq\alpha} = e^{\left(\frac{1}{2}\sigma^{ab}\theta\_{ab} + iq\alpha\mathbb{I}\right)} = \mathbb{S}$$

as the Lorentz-phase transformation in its most complete form possible. This form is also called spinorial transformation. It is what we will employ to define the spinorial fields *ψ* as the column of four complex functions that are scalars for coordinate transformations while transforming according to *ψ* ′ =*Sψ* under the spinorial transformations.

We may now summarize by saying that, given the most general spinorial transformation *S*, the column and row of complex scalars *ψ* and *ψ* transforming as

$$
\psi' = \mathbf{S}\psi \qquad \qquad \overline{\psi}' = \overline{\psi}\mathbf{S}^{-1} \tag{51}
$$

are called *spinorial fields*. Operations of sum and product respect spinor transformation.

As above, the transformation *S* is local and so we have to introduce the spinorial connection defined in terms of the transformation law that guarantees the derivative to be covariant for general spinorial transformations.

Therefore, we have that the coefficients **Ω***<sup>ν</sup>* transforming according to

$$\boldsymbol{\Omega}'\_{\boldsymbol{\nu}} = \mathbf{S} \left( \boldsymbol{\Omega}\_{\boldsymbol{\nu}} - \mathbf{S}^{-1} \boldsymbol{\partial}\_{\boldsymbol{\nu}} \mathbf{S} \right) \mathbf{S}^{-1} \tag{52}$$

are called *spinorial connection*. Once the spinorial connection is assigned, we have that

$$\mathcal{D}\_{\mu}\psi = \partial\_{\mu}\psi + \mathbf{\Omega}\_{\mu}\psi \qquad \mathbf{D}\_{\mu}\overline{\psi} = \partial\_{\mu}\overline{\psi} - \overline{\psi}\mathbf{\Omega}\_{\mu} \tag{53}$$

are the *covariant derivatives* of the spinorial fields.

We now give a list of properties of the Clifford matrices.

We have that the *Clifford matrices γ a* such that

$$
\Lambda \gamma^b \Lambda^{-1} \equiv (\Lambda^{-1})^b\_a \gamma^a \tag{54}
$$

verify the anticommutation relationships

$$\{\gamma\_{a\prime}\gamma\_{b}\} = 2\eta\_{ab}\mathbb{I} \tag{55}$$

so that we can define the matrices *σab* as

$$
\sigma\_{ab} = \frac{1}{4} [\gamma\_{a\prime} \gamma\_{b\prime}] \tag{56}
$$

and

$$
\sigma\_{ab} = -\frac{i}{2} \varepsilon\_{abcd} \pi \sigma^{cd} \tag{57}
$$

for the *π* matrix to be implicitly defined. Then,

$$
\gamma\_0 \gamma\_a^\dagger \gamma\_0 = \gamma\_a \tag{58}
$$

$$
\gamma\_0 \sigma\_{ab}^\dagger \gamma\_0 = -\sigma\_{ab} \tag{59}
$$

$$
\boldsymbol{\pi}^{\dagger} = \boldsymbol{\pi} \tag{60}
$$

alongside the square properties

$$
\gamma\_{\mathfrak{a}} \gamma^{\mathfrak{a}} = 4 \mathbb{I} \tag{61}
$$

$$
\sigma\_{ab}\sigma^{ab} = -\Im \mathbb{I} \tag{62}
$$

$$
\pi^2 = \mathbb{I} \tag{63}
$$

together with the anticommutation properties

$$\{\pi, \gamma\_a\} = 0\tag{64}$$

$$\{\gamma\_{i\prime}\sigma\_{jk}\} = i\varepsilon\_{ijkq}\pi\gamma^{q} \tag{65}$$

and the commutation properties

$$[\pi, \sigma\_{ab}] = 0 \tag{66}$$

$$[\gamma\_{a\prime}\sigma\_{bc}] = \eta\_{ab}\gamma\_{c} - \eta\_{ac}\gamma\_{b} \tag{67}$$

$$
\begin{bmatrix}
\sigma\_{ab\prime}\sigma\_{cd}
\end{bmatrix} = \eta\_{ad}\sigma\_{bc} - \eta\_{ac}\sigma\_{bd} + \eta\_{bc}\sigma\_{ad} - \eta\_{bd}\sigma\_{ac} \tag{68}
$$

as well as

$$
\gamma\_a \gamma\_b = \eta\_{ab} \mathbb{I} + 2 \sigma\_{ab} \tag{69}
$$

$$
\gamma\_i \gamma\_j \gamma\_k = \gamma\_i \eta\_{jk} - \gamma\_j \eta\_{ik} + \gamma\_k \eta\_{ij} + i \varepsilon\_{ijkq} \pi \gamma^q \tag{70}
$$

all being spinorial identities. Employing *γ*0, we can define

$$
\overline{\psi} = \psi^\dagger \gamma\_0 \qquad \qquad \gamma\_0 \overline{\psi}^\dagger = \psi \tag{71}
$$

as the *spinor conjugation*. In particular, we have

$$
\pi\_L = \frac{1}{2}(\mathbb{I} - \pi) \tag{72}
$$

$$
\pi\_{\mathbb{R}} = \frac{1}{2} (\mathbb{I} + \pi) \tag{73}
$$

as *left-handed/right-handed chiral projectors*. They verify

$$
\pi\_L^\dagger = \pi\_L \tag{74}
$$

$$
\pi\_R^\dagger = \pi\_R \tag{75}
$$

alongside

$$
\pi\_L^2 = \pi\_L \tag{76}
$$

$$
\pi\_R^2 = \pi\_R \tag{77}
$$

together with

$$
\boldsymbol{\pi}\_L \boldsymbol{\pi}\_R = \boldsymbol{\pi}\_R \boldsymbol{\pi}\_L = 0 \tag{78}
$$

and such that

$$
\boldsymbol{\pi}\_L + \boldsymbol{\pi}\_R = \mathbb{I} \tag{79}
$$

in general. We can also define

$$
\pi\_L \psi = \psi\_L \qquad \quad \quad \overline{\psi} \pi\_R = \overline{\psi}\_L \tag{80}
$$

$$
\pi\_R \psi = \psi\_R \qquad \quad \quad \overline{\psi} \,\pi\_L = \overline{\psi}\_R \tag{81}
$$

and

$$
\overline{\psi}\_L + \overline{\psi}\_R = \overline{\psi} \qquad \psi\_L + \psi\_R = \psi \tag{82}
$$

as *left-handed/right-handed chiral parts*. With the pair of conjugate spinors, we define the *bi-linear spinorial quantities* according to


such that they are all real tensor quantities. From them,

$$\begin{aligned} \psi \overline{\psi} &\equiv \frac{1}{4} \Phi \mathbb{I} + \frac{1}{4} \mathcal{U}\_a \gamma^a + \frac{i}{8} \mathcal{M}\_{ab} \sigma^{ab} - \\\ &- \frac{1}{8} \Sigma\_{ab} \sigma^{ab} \pi - \frac{1}{4} S\_a \gamma^a \pi - \frac{i}{4} \Theta \pi \end{aligned} \tag{89}$$

from which we get the relationships

$$2\mathcal{U}\_{\mu}\mathcal{S}\_{\nu}\sigma^{\mu\nu}\,\pi\psi + \mathcal{U}^2\psi = 0\tag{90}$$

$$i\Theta S\_{\mu}\gamma^{\mu}\psi + \Phi S\_{\mu}\gamma^{\mu}\pi\psi + lI^2\psi = 0\tag{91}$$

and

$$
\Delta L\_a \gamma^a \psi = -S\_a \gamma^a \pi \psi = (\Phi \mathbb{I} + i \Theta \pi) \psi \tag{92}
$$

as well as the relationships

$$
\Sigma^{ab} = -\frac{1}{2} \varepsilon^{abij} M\_{ij} \tag{93}
$$

$$M^{ab} = \frac{1}{2} \varepsilon^{abij} \Sigma\_{ij} \tag{94}$$

and

$$M\_{ab}\Phi - \Sigma\_{ab}\Theta = \mathcal{U}^{\dot{J}}\mathcal{S}^{k}\varepsilon\_{jkab} \tag{95}$$

$$\mathcal{M}\_{ab}\Theta + \Sigma\_{ab}\Phi = \mathcal{U}\_{[a}\mathcal{S}\_{b]}\tag{96}$$

with

$$M\_{ik}\mathcal{U}^i = \Theta \mathcal{S}\_k \tag{97}$$

$$
\Sigma\_{ik}\mathcal{U}^{\dot{i}} = \Phi \mathcal{S}\_k \tag{98}
$$

$$M\_{ik}S^i = \Theta \mathcal{U}\_k \tag{99}$$

$$
\Sigma\_{ik} S^i = \Phi \mathcal{U} l\_k \tag{100}
$$

and also

$$-\frac{1}{2}M\_{ab}M^{ab} = -\frac{1}{2}\Sigma\_{ab}\Sigma^{ab} = \Phi^2 - \Theta^2\tag{101}$$

$$\mathcal{U}\_{a}\mathcal{U}^{a} = -\mathcal{S}\_{a}\mathcal{S}^{a} = \Theta^{2} + \Phi^{2} \tag{102}$$

$$\,\_{2}^{1}M\_{ab}\Sigma^{ab} = -2\Theta\Phi \tag{103}$$

$$\mathcal{U}\_{\mathfrak{a}}\mathcal{S}^{\mathfrak{a}} = 0 \tag{104}$$

called *Fierz re-arrangements* of spinor fields. If both scalars Θ and Φ do not vanish identically, we can always find a special frame where the most general spinor is written as

$$
\psi = \phi e^{-\frac{i}{2}\beta\pi} e^{-i\alpha} \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix} \tag{105}
$$

up to the reversal of the third axis and up to the discrete transformation *ψ*→*πψ* and called *polar form*. From this, we can write

$$
\Sigma^{ab} = 2\phi^2(\cos\beta u^{[a}s^{b]} - \sin\beta u\_j s\_k \varepsilon^{j kab}) \tag{106}
$$

$$M^{ab} = 2\phi^2(\cos\beta u\_j s\_k \varepsilon^{j k ab} + \sin\beta u^{[a} s^{b]}) \tag{107}$$

in terms of

$$S^{a} = \mathcal{Q}\phi^{2}s^{a} \tag{108}$$

$$\mathcal{U}^a = 2\phi^2 u^a \tag{109}$$

and

$$
\Theta = 2\phi^2 \sin \beta \tag{110}
$$

$$
\Phi = 2\phi^2 \cos \beta \tag{111}
$$

showing that the fields *φ* and *β* are a scalar and a pseudo-scalar, respectively. Then,

$$
\mu\_a \mathfrak{u}^a = -s\_a \mathfrak{s}^a = 1 \tag{112}
$$

$$
\mu\_a \mathbf{s}^a = \mathbf{0} \tag{113}
$$

showing that the normalized velocity vector *u <sup>a</sup>* and the normalized spin axial-vector *s a* possess three independent components each. This means that *φ* and *β* are the only true real scalar degrees of freedom and called *module* and *Yvon–Takabayashi angle*. The reader interested in details for all these statements can have a look at [15].

The conditions of compatibility now read *Dµγ<sup>a</sup>* =0 in general: if the spinorial matrix also has a tensorial index, the covariant derivative is to be completed to the form

$$\mathbf{D}\_{\mu}\mathbf{B}\_{a} = \partial\_{\mu}\mathbf{B}\_{a} - \mathbf{B}\_{b}\Omega^{b}\_{\phantom{a}a\mu} + [\mathbf{D}\_{\mu\nu}\mathbf{B}\_{a}]^{\phantom{a}}$$

which can be taken for the gamma matrix and hence implementing the above condition, and recalling that these matrices in Lorentz indices are constants, yields

$$-\gamma\_b \Omega^b{}\_{a\mu} + [\Omega\_{\mu\nu}\gamma\_a] = 0$$

as a relation among connections. By writing a general

$$
\Omega\_{\mu} = a \Omega^{i\bar{j}}{}\_{\mu} \sigma\_{i\bar{j}} + A\_{\mu}
$$

and plugging it into the above relation, we obtain that

$$-\gamma\_b \Omega^b{}\_{k\mu} + a \Omega^{ij}{}\_{\mu} [\sigma\_{ij\prime} \gamma\_k] + [A\_{\mu\prime} \gamma\_k] = 0$$

and with [*σij*, *γ<sup>k</sup>* ] = *ηkjγi*−*ηkiγ<sup>j</sup>* we get *a*=1/2 and

$$[A\_{\mu\nu}\gamma\_s] = 0$$

telling that *A<sup>µ</sup>* must commute with all gamma matrices, and thus, with all possible matrices, implying that it must be proportional to the identity matrix. Writing it as

$$\mathbf{A}\_{\mu} = (p\mathbf{C}\_{\mu} + ibA\_{\mu})\mathbb{T}\_{\star}$$

it is possible to see that, for *b* = *q*, it is possible to interpret the vector *A<sup>µ</sup>* as the gauge potential. Because the other term is related to conformal transformations, which are not symmetries in our case, we set *p*=0 in general. Then, we have that, all considered, we may write the expression

$$\mathbf{\Omega}\_{\mu} = \frac{1}{2} \boldsymbol{\Omega}^{\ddot{ij}}{}\_{\mu} \sigma\_{\dot{ij}} + iqA\_{\mu}\boldsymbol{\Gamma}$$

as the most general form of spinorial connection.

To summarize, we have that the most general spinorial connection is given by

$$
\Omega\_{\mu} = \frac{1}{2} \Omega\_{ab\mu} \sigma^{ab} + iqA\_{\mu}\mathbb{I} \tag{114}
$$

in terms of the generator-valued spin connection and the gauge potential, and this is equivalent to the fact that the spinorial covariant derivatives of the gamma matrices are

$$D\_{\mu}\gamma\_{a} = 0\tag{115}$$

vanishing identically, as it is quite straightforward to see.

We have that, from the spinorial connection, we define

$$F\_{a\beta} = \partial\_a \mathbf{\Omega}\_{\beta} - \partial\_{\beta} \mathbf{\Omega}\_a + [\mathbf{\Omega}\_{a\prime} \mathbf{\Omega}\_{\beta}] \tag{116}$$

as the *spinorial curvature*. With it,

$$[\mathbf{D}\_{\mu\nu}\mathbf{D}\_{\nu}]\psi = \mathcal{Q}^{\mu}{}\_{\mu\nu}\mathbf{D}\_{a}\psi + \mathbf{F}\_{\mu\nu}\psi\tag{117}$$

as *commutator of covariant derivatives* of spinor fields. Correspondingly, the curvature is decomposable as

$$F\_{\mu\nu} = \frac{1}{2} G\_{ab\mu\nu} \sigma^{ab} + iqF\_{\mu\nu}\mathbb{I} \tag{118}$$

with the curvature tensor and gauge strength. For a final step, we have

$$D\_{\mu}\mathbf{F}\_{\mathbf{x}\rho} + D\_{\mathbf{x}}\mathbf{F}\_{\rho\mu} + D\_{\rho}\mathbf{F}\_{\mu\kappa} + \\
$$

$$-\mathbf{F}\_{\beta\mu}\mathbf{Q}^{\beta}\_{\ \rho\kappa} + \mathbf{F}\_{\beta\kappa}\mathbf{Q}^{\beta}\_{\ \mu\rho} + \mathbf{F}\_{\beta\rho}\mathbf{Q}^{\beta}\_{\ \kappa\mu} \equiv \mathbf{0} \tag{119}$$

as spinorial geometrical identities holding in general.

We conclude with some fundamental comments: the first and most important one is about the fact that so far we have encountered three types of transformation laws: the first type was the most general coordinate transformation; the second type was the gauge transformation; the third type was the specific Lorentz transformation, which was given in real representation for tensors and complex representation for spinors. The coordinate transformation is known as *passive transformation*; the Lorentz transformation in real representation as well as the Lorentz transformation in complex representation merged with the gauge transformation that is the spinor transformation, are known altogether as *active transformations*. Because they have the very same parameters, we then have that both active transformations have to be performed simultaneously.

Another interesting comment is on the connections and how they are built: the torsion tensor, when the metric tensor is used, gives the connection (16); this connection, when the dual bases of tetrad fields are employed, gives the spin connection (37); this spin connection, when the gamma matrices and their commutators are considered, with the gauge potential, when multiplied by the identity matrix, give the spinorial connection (114). Remarkably, all fields fit within the most general spinorial connection, with no room for anything else: this circumstance can be seen as a sort of geometric unification of all the physical fields that are involved. On the other hand, however, in order to see it that way, we have to wait until we interpret these geometric quantities.

A final comment regards the structure of the covariant commutator (117), in which, by interpreting the covariant derivative as the covariant generators of translations, one sees that the completely antisymmetric torsion plays the role that in Lie group theory is played by the completely antisymmetric structure coefficients; we also recall to the reader that, in the curvature, there appear sigma matrices which are the generators of the Lorentz transformations and therefore of the space–time rotations. An additional interpretation that can be assigned to the covariant commutator is that, when a field is moved around, it would fail to go back to the starting point and have the initial orientation. A position mismatch is measured by torsion and a directional mismatch is measured by curvature, and this is why torsion is also said to describe the dislocations while curvature is also said to describe the disclinations of a round trip. This shows intuitively that both torsion and curvature have to be accounted for the most general description of space–time.

For some introduction to the general theory of spinors and their classifications, we refer the readers to [16,17].

### *Geometry and Matter in Interaction*

Now that, in terms of general symmetry arguments, we have completed the definitions of all geometric quantities for the kinematic background, the next step is to have them coupled to one another in order to assign their dynamics.
