Is This Expression From Dirac Field Wrong In P&S?

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Introduction

Quantum Field Theory (QFT) is a fundamental framework in modern physics that describes the behavior of fundamental particles and forces in the universe. The Dirac Equation, introduced by Paul Dirac in 1928, is a key component of QFT, describing the behavior of fermions, which are particles that follow Fermi-Dirac statistics. In the context of QFT, the Dirac Equation is used to describe the behavior of fermions in the presence of an electromagnetic field. In this article, we will discuss a specific expression from Peskin and Schroeder's book on QFT, which attempts to demonstrate why quantizing with the commutator is wrong.

Background

In QFT, the Dirac Equation is used to describe the behavior of fermions in the presence of an electromagnetic field. The Dirac Equation is a relativistic wave equation that describes the behavior of fermions in terms of their wave functions. The wave functions of fermions are described by a set of four-component spinors, which are mathematical objects that describe the spin and momentum of the fermion. The Dirac Equation is a fundamental equation in QFT, and it is used to describe the behavior of fermions in a wide range of physical systems.

The Problem with Quantizing with the Commutator

In QFT, the commutator is used to describe the behavior of bosons, which are particles that follow Bose-Einstein statistics. However, the commutator is not suitable for describing the behavior of fermions, which follow Fermi-Dirac statistics. In particular, the commutator does not satisfy the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state simultaneously. This is a fundamental property of fermions, and it is not satisfied by the commutator.

In equation 3.87 of Peskin and Schroeder's book on QFT, the authors attempt to demonstrate why quantizing with the commutator is wrong. The expression in question is:

[ψ(x),ψ†(y)]=∫d3p(2π)312Ep[u(p)uˉ(p)]e−ip(x−y)\left[ \psi(x), \psi^\dagger(y) \right] = \int \frac{d^3p}{(2\pi)^3} \frac{1}{2E_p} \left[ u(p) \bar{u}(p) \right] e^{-ip(x-y)}

where ψ(x)\psi(x) and ψ†(y)\psi^\dagger(y) are the fermion field operators, u(p)u(p) and uˉ(p)\bar{u}(p) are the spinors that describe the fermion, and EpE_p is the energy of the fermion.

The Issue with the Expression

The issue with the expression in equation 3.87 is that it does not satisfy the Pauli exclusion principle. The commutator on the left-hand side of the equation does not vanish when the two fermions are in the same quantum state, which is a fundamental property of fermions. This is because the commutator does not take into account the spin and momentum of the fermions, which are essential for describing the behavior of fermions.

A Correct Expression

A correct expression for the commutator of the fermion field operators is:

\left[ \psi(x), \psi^\dagger(y) \] = \int \frac{d^3p}{(2\pi)^3} \frac{1}{2E_p} \left[ u(p) \bar{u}(p) \right] e^{-ip(x-y)} \left( 1 - \frac{1}{2} \left[ u(p) \bar{u}(p) \right] \right)

where the additional term (1−12[u(p)uˉ(p)])\left( 1 - \frac{1}{2} \left[ u(p) \bar{u}(p) \right] \right) takes into account the spin and momentum of the fermions, and ensures that the commutator satisfies the Pauli exclusion principle.

Conclusion

In conclusion, the expression in equation 3.87 of Peskin and Schroeder's book on QFT is incorrect because it does not satisfy the Pauli exclusion principle. A correct expression for the commutator of the fermion field operators is given by the expression above, which takes into account the spin and momentum of the fermions. This expression is essential for describing the behavior of fermions in QFT, and it is a fundamental component of the Dirac Equation.

References

  • Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Addison-Wesley.
  • Dirac, P. A. M. (1928). The quantum theory of the electron. Proceedings of the Royal Society of London A, 117(778), 610-624.

Further Reading

For further reading on QFT and the Dirac Equation, we recommend the following resources:

  • "Quantum Field Theory for the Gifted Amateur" by Tom Lancaster and Stephen J. Blundell
  • "The Quantum Theory of Fields" by Steven Weinberg
  • "Quantum Field Theory and the Standard Model" by Michael E. Peskin and Daniel V. Schroeder

Introduction

In our previous article, we discussed a specific expression from Peskin and Schroeder's book on Quantum Field Theory (QFT), which attempts to demonstrate why quantizing with the commutator is wrong. We also provided a correct expression for the commutator of the fermion field operators. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is the Dirac Equation?

A: The Dirac Equation is a relativistic wave equation that describes the behavior of fermions, which are particles that follow Fermi-Dirac statistics. The Dirac Equation is a fundamental equation in QFT, and it is used to describe the behavior of fermions in a wide range of physical systems.

Q: Why is the commutator not suitable for describing the behavior of fermions?

A: The commutator is not suitable for describing the behavior of fermions because it does not satisfy the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state simultaneously. This is a fundamental property of fermions, and it is not satisfied by the commutator.

Q: What is the correct expression for the commutator of the fermion field operators?

A: The correct expression for the commutator of the fermion field operators is:

[ψ(x),ψ†(y)]=∫d3p(2π)312Ep[u(p)uˉ(p)]e−ip(x−y)(1−12[u(p)uˉ(p)])\left[ \psi(x), \psi^\dagger(y) \right] = \int \frac{d^3p}{(2\pi)^3} \frac{1}{2E_p} \left[ u(p) \bar{u}(p) \right] e^{-ip(x-y)} \left( 1 - \frac{1}{2} \left[ u(p) \bar{u}(p) \right] \right)

where the additional term (1−12[u(p)uˉ(p)])\left( 1 - \frac{1}{2} \left[ u(p) \bar{u}(p) \right] \right) takes into account the spin and momentum of the fermions, and ensures that the commutator satisfies the Pauli exclusion principle.

Q: Why is the expression in equation 3.87 of Peskin and Schroeder's book on QFT incorrect?

A: The expression in equation 3.87 of Peskin and Schroeder's book on QFT is incorrect because it does not satisfy the Pauli exclusion principle. The commutator on the left-hand side of the equation does not vanish when the two fermions are in the same quantum state, which is a fundamental property of fermions.

Q: What are the implications of this result?

A: The implications of this result are that the commutator is not a suitable operator for describing the behavior of fermions in QFT. Instead, the correct operator is the one that satisfies the Pauli exclusion principle, which is the expression we provided above.

Q: What are some other resources that can help me learn more about QFT and the Dirac Equation?

A: Some other resources that can help you learn more about QFT and the Dirac Equation include:

  • "Quantum Field Theory for the Gifted Amateur" by Tom Lancaster and Stephen J. Blundell
  • "The Quantum Theory of Fields" by Steven Weinberg
  • "Quantum Field Theory and the Standard Model" by Michael E. Peskin and Daniel V. Schroeder

Conclusion

In conclusion, the expression in equation 3.87 of Peskin and Schroeder's book on QFT is incorrect because it does not satisfy the Pauli exclusion principle. A correct expression for the commutator of the fermion field operators is provided above, which takes into account the spin and momentum of the fermions, and ensures that the commutator satisfies the Pauli exclusion principle. We hope that this Q&A article has been helpful in clarifying some of the key concepts related to this topic.

References

  • Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Addison-Wesley.
  • Dirac, P. A. M. (1928). The quantum theory of the electron. Proceedings of the Royal Society of London A, 117(778), 610-624.

Further Reading

For further reading on QFT and the Dirac Equation, we recommend the following resources:

  • "Quantum Field Theory for the Gifted Amateur" by Tom Lancaster and Stephen J. Blundell
  • "The Quantum Theory of Fields" by Steven Weinberg
  • "Quantum Field Theory and the Standard Model" by Michael E. Peskin and Daniel V. Schroeder