String Theory

Covariant form and relativistic invariance

To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:

$\gamma^0 = \beta \,$

$\gamma^k = \gamma^0 \alpha^k. \,$

and the equation takes the form

$i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0 \,.$

In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is

$\gamma^0 = \left(\begin{array}{cccc} I_2 & 0 \\ 0 & -I_2 \end{array}\right), \gamma^1 = \left(\begin{array}{cccc} 0 & \sigma_x \\ -\sigma_x & 0 \end{array}\right), \gamma^2 = \left(\begin{array}{cccc} 0 & \sigma_y \\ -\sigma_y & 0 \end{array}\right), \gamma^3 = \left(\begin{array}{cccc} 0 & \sigma_z \\ -\sigma_z & 0 \end{array}\right). \,$

The complete system is summarized using the Minkowski metric on spacetime in the form

$\{\gamma^\mu,\gamma^\nu\} = 2 g^{\mu\nu} \,$

where the bracket expression

$\{a, b\} = ab + ba$

denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-d space with metric signature

(+ - - -)

. The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.

The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:

$P_\mathrm{op}\psi = mc\psi. \,$

Using ${\partial\!\!\!\big /}$ (pronounced: "d-slash") in Feynman slash notation, which includes the gamma matrices as well as a summation over the spinor components in the derivative itself, the Dirac equation becomes:

$i \hbar {\partial\!\!\!\big /} \psi - m c \psi = 0$

In practice, physicists often use units of measure such that

ħ = c = 1

, known as natural units. The equation then takes the simple form

$(i{\partial\!\!\!\big /} - m) \psi = 0\,$

In the limit $m \rightarrow 0$ , the Dirac equation reduces to the Weyl equation, which describes massless spin-1/2 particles.

A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:

$\gamma^{\mu\prime} = S^{-1} \gamma^\mu S.$

If in addition the matrices are all unitary, as are the Dirac set, then

S

itself is unitary;

$\gamma^{\mu\prime} = U^\dagger \gamma^\mu U.$

The transformation

U

is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator

γ μ \partial μ

to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form

$( iU^\dagger \gamma^\mu U\partial_\mu^\prime - m)\psi(x^\prime,t^\prime) = 0$

$U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi(x^\prime,t^\prime) = 0.$

If we now define the transformed spinor

$\psi^\prime = U\psi$

then we have the transformed Dirac equation in a way that demonstrates manifest relativistic invariance:

$(i\gamma^\mu\partial_\mu^\prime - m)\psi^\prime(x^\prime,t^\prime) = 0.$

Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation. The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function (see below). The representation shown here is known as the standard representation - in it, the wave function's upper two components go over into Pauli's 2-spinor wave function in the limit of low energies and small velocities in comparison to light.

The considerations above reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation - they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as

γ μ γ ν

represent oriented surface elements, and so on. With this in mind, we can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is

$V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta.$

For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of

\sqrt g

, where

g

is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus

$V = i \gamma^0\gamma^1\gamma^2\gamma^3.\$

This matrix is given the special symbol

γ 5

, owing to its importance when one is considering improper transformations of spacetime, that is, those that change the orientation of the basis vectors. In the standard representation it is

$\gamma_5 = \begin{pmatrix} 0 & I_{2} \\ I_{2} & 0 \end{pmatrix}.$

This matrix will also be found to anticommute with the other four Dirac matrices. It takes a leading role when questions of parity arise, because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on space-time.

String Theory

Friday, 7 June 2013

Covariant form and relativistic invariance

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