Do Lorentz Transformed Gamma Matrices Satisfy The Anticommutation Rule?

Introduction

In the realm of special relativity and quantum mechanics, the Dirac equation plays a pivotal role in describing the behavior of fermions. The Dirac equation is a relativistic wave equation that incorporates the principles of quantum mechanics and special relativity. One of the key components of the Dirac equation is the gamma matrix, which is a set of four 4x4 matrices that satisfy a specific anticommutation relation. In this article, we will delve into the question of whether Lorentz transformed gamma matrices satisfy the anticommutation rule.

Notation

Before we proceed, let's establish the notation used in this article. The gamma matrices are denoted by $\gamma^{\mu}$, where $\mu$ represents the four spacetime dimensions. The metric tensor is denoted by $\eta^{\mu\nu}$, and the Minkowski metric is given by:

$$\eta^{\mu\nu} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$$

The Lorentz transformation matrix is denoted by $\Lambda^{\mu}_{\nu}$, and the inverse Lorentz transformation matrix is denoted by $(\Lambda^{-1})^{\mu}_{\nu}$.

Context

The context of this discussion is the proof of relativistic invariance of the Dirac equation presented by Bethe and Jackiw [1]. In their proof, they write that the Dirac equation is invariant under Lorentz transformations, which implies that the gamma matrices must satisfy a specific relation under Lorentz transformations. Specifically, they write that the Lorentz transformed gamma matrices must satisfy the anticommutation relation:

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

Lorentz Transformation of Gamma Matrices

To investigate whether the Lorentz transformed gamma matrices satisfy the anticommutation rule, we need to first determine the form of the Lorentz transformed gamma matrices. The Lorentz transformation of the gamma matrices is given by:

$$\gamma^{\prime\mu} = \Lambda^{\mu}_{\nu} \gamma^{\nu}$$

Using the definition of the Lorentz transformation matrix, we can rewrite the above equation as:

$$\gamma^{\prime\mu} = \sum_{\nu} \Lambda^{\mu}_{\nu} \gamma^{\nu}$$

Anticommutation Relation

The anticommutation relation is given by:

$$\{\gamma^{\mu}, \gamma^{\nu}\} = \gamma^{\mu} \gamma^{\nu} + \gamma^{\nu} \gamma^{\mu}$$

Using the definition of the Lorentz transformed gamma matrices, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \gamma^{\prime\mu} \gamma^{\prime\nu} + \gamma^{\prime\nu} \gamma^{\prime\mu}$$

Proof of Anticommutation Relation

To prove that the Lorentz transformed gamma matrices satisfy the anticommutation relation, we need to show that:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\eta^{\prime\mu\nu}$$

where $\eta^{\prime\mu\nu}$ is the Minkowski metric in the primed frame.

Using the definition of the Lorentz transformed gamma matrices, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \{\gamma^{\alpha}, \gamma^{\beta}\}$$

Using the anticommutation relation, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} (2\eta^{\alpha\beta})$$

Using the definition of the Minkowski metric, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \eta^{\alpha\beta}$$

Using the definition of the Lorentz transformation matrix, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\eta^{\prime\mu\nu}$$

Conclusion

In conclusion, we have shown that the Lorentz transformed gamma matrices satisfy the anticommutation relation. This result is crucial for the proof of relativistic invariance of the Dirac equation, as it ensures that the Dirac equation is invariant under Lorentz transformations.

References

[1] Bethe, H. A., & Jackiw, R. (1968). Intermediate Quantum Mechanics. W. A. Benjamin.

[0] Dirac, P. A. M. (1928). The quantum theory of the electron. Proceedings of the Royal Society of London A, 117(778), 610-624.

Appendix

In this appendix, we provide a brief overview of the Dirac equation and its relation to the gamma matrices.

The Dirac equation is a relativistic wave equation that describes the behavior of fermions. It is given by:

$$i\hbar \frac{\partial \psi}{\partial t} = (c\alpha \cdot \mathbf{p} + \beta m c^2) \psi$$

where $\psi$ is the wave function, $\alpha$ and $\beta$ are the Dirac matrices, $\mathbf{p}$ is the momentum operator, and $m$ is the mass of the fermion.

The Dirac matrices are given by:

$$\alpha^i = \begin{pmatrix} 0 & \sigma^i \\ \sigma^i & 0 \end{pmatrix}$$

$$\beta = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}$$

where $\sigma^i$ are the Pauli matrices and $I$ is the identity matrix.

The gamma matrices are related to the Dirac matrices by:

$$\gamma^{\mu} = \begin{pmatrix} 0 & \sigma^{\mu} \\ \sigma^{\mu} & 0 \end{pmatrix}$$

where $\sigma^{\mu}$ are the gamma matrices.

The anticommutation relation is given by:

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

where $\eta^{\mu\nu}$ is the Minkowski metric.

The Lorentz transformation of the gamma matrices is given by:

$$\gamma^{\prime\mu} = \Lambda^{\mu}_{\nu} \gamma^{\nu}$$

Using the definition of the Lorentz transformation matrix, we can rewrite the above equation as:

$$\gamma^{\prime\mu} = \sum_{\nu} \Lambda^{\mu}_{\nu} \gamma^{\nu}$$

The anticommutation relation is given by:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \gamma^{\prime\mu} \gamma^{\prime\nu} + \gamma^{\prime\nu} \gamma^{\prime\mu}$$

Using the definition of the Lorentz transformed gamma matrices, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \{\gamma^{\alpha}, \gamma^{\beta}\}$$

Using the anticommutation relation, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} (2\eta^{\alpha\beta})$$

Using the definition of the Minkowski metric, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \eta^{\alpha\beta}$$

Using the definition of the Lorentz transformation matrix, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\eta^{\prime\mu\nu}$$

Q: What is the significance of the anticommutation rule in the context of the Dirac equation?

A: The anticommutation rule is a fundamental property of the gamma matrices that ensures the Dirac equation is invariant under Lorentz transformations. It is a crucial component of the Dirac equation and plays a key role in its relativistic invariance.

Q: What is the relationship between the gamma matrices and the Dirac matrices?

A: The gamma matrices are related to the Dirac matrices by:

$$\gamma^{\mu} = \begin{pmatrix} 0 & \sigma^{\mu} \\ \sigma^{\mu} & 0 \end{pmatrix}$$

where $\sigma^{\mu}$ are the gamma matrices.

Q: How do the Lorentz transformed gamma matrices satisfy the anticommutation rule?

A: The Lorentz transformed gamma matrices satisfy the anticommutation rule due to the following relation:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \{\gamma^{\alpha}, \gamma^{\beta}\}$$

Using the anticommutation relation, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} (2\eta^{\alpha\beta})$$

Using the definition of the Minkowski metric, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \eta^{\alpha\beta}$$

Using the definition of the Lorentz transformation matrix, we can rewrite the above equation as:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = 2\eta^{\prime\mu\nu}$$

Q: What is the implication of the anticommutation rule on the Dirac equation?

A: The anticommutation rule ensures that the Dirac equation is invariant under Lorentz transformations. This means that the Dirac equation remains the same in all inertial frames of reference, which is a fundamental requirement of special relativity.

Q: Can you provide an example of how the anticommutation rule is used in the Dirac equation?

A: Yes, the anticommutation rule is used in the Dirac equation to ensure that the wave function is invariant under Lorentz transformations. For example, consider the Dirac equation in the primed frame:

i\hbar \frac{\partial \psi^{\prime}}{\partial t^{\prime}} = (c\alpha \cdot \mathbf{p}^{\prime} + \beta m c^2) \psi^{\}

Using the anticommutation rule, we can rewrite the above equation as:

$$i\hbar \frac{\partial \psi^{\prime}}{\partial t^{\prime}} = (c\alpha \cdot \mathbf{p}^{\prime} + \beta m c^2) \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \psi^{\alpha}$$

Using the definition of the Lorentz transformation matrix, we can rewrite the above equation as:

$$i\hbar \frac{\partial \psi^{\prime}}{\partial t^{\prime}} = (c\alpha \cdot \mathbf{p}^{\prime} + \beta m c^2) \psi^{\prime}$$

This shows that the Dirac equation is invariant under Lorentz transformations, which is a direct consequence of the anticommutation rule.

Q: What are the implications of the anticommutation rule on the physical interpretation of the Dirac equation?

A: The anticommutation rule has significant implications for the physical interpretation of the Dirac equation. It ensures that the wave function is a relativistic wave function, which means that it satisfies the principles of special relativity. This has important consequences for the physical interpretation of the Dirac equation, including the prediction of antiparticles and the existence of wave-particle duality.

Q: Can you provide a summary of the key points discussed in this article?

A: Yes, the key points discussed in this article are:

• The anticommutation rule is a fundamental property of the gamma matrices that ensures the Dirac equation is invariant under Lorentz transformations.
• The Lorentz transformed gamma matrices satisfy the anticommutation rule due to the relation:

$$\{\gamma^{\prime\mu}, \gamma^{\prime\nu}\} = \sum_{\alpha} \sum_{\beta} \Lambda^{\mu}_{\alpha} \Lambda^{\nu}_{\beta} \{\gamma^{\alpha}, \gamma^{\beta}\}$$

• The anticommutation rule ensures that the Dirac equation is invariant under Lorentz transformations, which is a fundamental requirement of special relativity.
• The anticommutation rule has significant implications for the physical interpretation of the Dirac equation, including the prediction of antiparticles and the existence of wave-particle duality.