Do Lorentz Transformed Gamma Matrices Satisfy The Anticommutation Rule?

Introduction

In the realm of quantum mechanics and quantum field theory, the Dirac equation plays a pivotal role in describing the behavior of fermions, particularly electrons. The Dirac equation is a relativistic wave equation that incorporates the principles of special relativity and quantum mechanics. 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 (0, 1, 2, and 3). The metric tensor is denoted by $\eta_{\mu\nu}$, and the Minkowski metric is given by $\eta_{\mu\nu} = \text{diag}(1, -1, -1, -1)$. 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 claim that the Lorentz transformed gamma matrices satisfy the anticommutation rule, which is a fundamental property of the gamma matrices.

The Anticommutation Rule

The anticommutation rule for the gamma matrices is given by:

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

where $\{\cdot, \cdot\}$ represents the anticommutator. This rule is a fundamental property of the gamma matrices and is used extensively in quantum field theory.

Lorentz Transformation of Gamma Matrices

Under a Lorentz transformation, the gamma matrices transform as:

$$\gamma^\mu \rightarrow \Lambda^\mu_\nu \gamma^\nu$$

where $\Lambda^\mu_\nu$ is the Lorentz transformation matrix. To determine whether the Lorentz transformed gamma matrices satisfy the anticommutation rule, we need to compute the anticommutator of the transformed gamma matrices.

Computing the Anticommutator

Using the definition of the anticommutator, we can write:

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

Under a Lorentz transformation, this becomes:

$$\{\gamma^\mu, \gamma^\nu\} = \Lambda^\mu_\alpha \Lambda^\nu_\beta \gamma^\alpha \gamma^\beta + \Lambda^\nu_\alpha \Lambda^\mu_\beta \gamma^\alpha \gamma^\beta$$

Using the properties of the Lorentz transformation matrix, can simplify this expression to:

$$\{\gamma^\mu, \gamma^\nu\} = \Lambda^\mu_\alpha \Lambda^\nu_\beta \{\gamma^\alpha, \gamma^\beta\}$$

Simplifying the Expression

Using the definition of the anticommutator, we can write:

$$\{\gamma^\alpha, \gamma^\beta\} = 2\eta^{\alpha\beta}$$

Substituting this into the previous expression, we get:

$$\{\gamma^\mu, \gamma^\nu\} = \Lambda^\mu_\alpha \Lambda^\nu_\beta 2\eta^{\alpha\beta}$$

Using the properties of the metric tensor, we can simplify this expression to:

$$\{\gamma^\mu, \gamma^\nu\} = 2\Lambda^\mu_\alpha \Lambda^\nu_\beta \eta^{\alpha\beta}$$

Evaluating the Expression

Using the definition of the Lorentz transformation matrix, we can write:

$$\Lambda^\mu_\alpha \Lambda^\nu_\beta \eta^{\alpha\beta} = \eta^{\mu\nu}$$

Substituting this into the previous expression, we get:

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

Conclusion

In conclusion, we have shown that the Lorentz transformed gamma matrices satisfy the anticommutation rule. This result is a fundamental property of the gamma matrices and is used extensively in quantum field theory. The proof of relativistic invariance of the Dirac equation presented by Bethe and Jackiw [1] relies on this result, and our derivation provides a clear and concise demonstration of the anticommutation rule for Lorentz transformed gamma matrices.

References

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

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

Appendix

In this appendix, we provide a brief derivation of the Lorentz transformation matrix. The Lorentz transformation matrix is given by:

$$\Lambda^\mu_\nu = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

where $\gamma = \frac{1}{\sqrt{1-\beta^2}}$ and $\beta = \frac{v}{c}$. This matrix satisfies the following properties:

$$\Lambda^\mu_\nu \Lambda^\nu_\rho = \delta^\mu_\rho$$

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Introduction

In our previous article, we explored the question of whether Lorentz transformed gamma matrices satisfy the anticommutation rule. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q: What is the anticommutation rule for gamma matrices?

A: The anticommutation rule for gamma matrices is given by:

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

where $\{\cdot, \cdot\}$ represents the anticommutator and $\eta^{\mu\nu}$ is the metric tensor.

Q: Why is the anticommutation rule important?

A: The anticommutation rule is a fundamental property of the gamma matrices and is used extensively in quantum field theory. It is essential for the derivation of the Dirac equation and the proof of relativistic invariance.

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

A: Under a Lorentz transformation, the gamma matrices transform as:

$$\gamma^\mu \rightarrow \Lambda^\mu_\nu \gamma^\nu$$

Using the definition of the anticommutator, we can write:

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

Substituting the transformed gamma matrices, we get:

$$\{\gamma^\mu, \gamma^\nu\} = \Lambda^\mu_\alpha \Lambda^\nu_\beta \gamma^\alpha \gamma^\beta + \Lambda^\nu_\alpha \Lambda^\mu_\beta \gamma^\alpha \gamma^\beta$$

Using the properties of the Lorentz transformation matrix, we can simplify this expression to:

$$\{\gamma^\mu, \gamma^\nu\} = \Lambda^\mu_\alpha \Lambda^\nu_\beta \{\gamma^\alpha, \gamma^\beta\}$$

Substituting the definition of the anticommutator, we get:

$$\{\gamma^\mu, \gamma^\nu\} = \Lambda^\mu_\alpha \Lambda^\nu_\beta 2\eta^{\alpha\beta}$$

Using the properties of the metric tensor, we can simplify this expression to:

$$\{\gamma^\mu, \gamma^\nu\} = 2\Lambda^\mu_\alpha \Lambda^\nu_\beta \eta^{\alpha\beta}$$

Using the definition of the Lorentz transformation matrix, we can write:

$$\Lambda^\mu_\alpha \Lambda^\nu_\beta \eta^{\alpha\beta} = \eta^{\mu\nu}$$

Substituting this into the previous expression, we get:

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

Q: What is the significance of the Lorentz transformation matrix?

A: The Lorentz transformation matrix is a fundamental concept in special relativity and is used to describe the transformation of spacetime coordinates under a Lorentz transformation. It is essential for the derivation of the Dirac equation and the proof of relativistic invariance.

Q: How does the Lorentz transformation matrix relate to the gamma matrices?

A: The Lorentz transformation matrix is used to transform the gamma matrices under a Lorentz transformation. Specifically, the gamma matrices transform as:

$$\gamma^\mu \rightarrow \Lambda^\mu_\nu \gamma^\nu$$

Using the properties of the Lorentz transformation matrix, we can simplify the expression for the anticommutator of the transformed gamma matrices.

Q: What is the relationship between the anticommutation rule and the Dirac equation?

A: The anticommutation rule is a fundamental property of the gamma matrices and is used extensively in the derivation of the Dirac equation. The Dirac equation is a relativistic wave equation that describes the behavior of fermions, and the anticommutation rule is essential for its derivation.

Q: What are the implications of the anticommutation rule for Lorentz transformed gamma matrices?

A: The anticommutation rule for Lorentz transformed gamma matrices has significant implications for the derivation of the Dirac equation and the proof of relativistic invariance. It ensures that the Dirac equation is invariant under Lorentz transformations, which is a fundamental property of the equation.

Conclusion

In conclusion, the anticommutation rule for Lorentz transformed gamma matrices is a fundamental property of the gamma matrices and is used extensively in quantum field theory. The Lorentz transformation matrix plays a crucial role in the derivation of the Dirac equation and the proof of relativistic invariance. The anticommutation rule ensures that the Dirac equation is invariant under Lorentz transformations, which is a fundamental property of the equation.