Dirac Propagator In Peskin & Schroeder's Book

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Introduction

In the realm of Quantum Field Theory (QFT), the Dirac propagator plays a crucial role in understanding the behavior of fermions, particularly in the context of the Dirac equation. The Dirac equation, introduced by Paul Dirac in 1928, is a relativistic wave equation that describes the behavior of fermions, such as electrons and quarks. In the book "An Introduction to Quantum Field Theory" by Peskin and Schroeder, the Dirac propagator is introduced as a fundamental concept in the study of QFT. In this article, we will delve into the Dirac propagator, its significance, and how it is derived in Peskin and Schroeder's book.

Notation and Dirac Equation

Before we dive into the Dirac propagator, let's establish some notation and review the Dirac equation. The Dirac equation is a relativistic wave equation that describes the behavior of fermions. It is given by:

iℏ∂ψ∂t=[cα⋅p+βmc2]ψi\hbar\frac{\partial\psi}{\partial t} = \left[c\alpha\cdot\mathbf{p} + \beta mc^2\right]\psi

where ψ\psi is the wave function of the fermion, α\alpha and β\beta are the Dirac matrices, cc is the speed of light, mm is the mass of the fermion, and p\mathbf{p} is the momentum of the fermion.

The Dirac matrices α\alpha and β\beta are given by:

αi=(0σiσi0),β=(I00−I)\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}

where σi\sigma_i are the Pauli matrices and II is the identity matrix.

Dirac Propagator

The Dirac propagator is a fundamental concept in QFT that describes the behavior of fermions in the presence of an external field. It is defined as the expectation value of the time-ordered product of two Dirac fields:

SF(x−y)=⟨0∣T[ψ(x)ψ‾(y)]∣0⟩S_F(x-y) = \left\langle 0 \right| T\left[ \psi(x) \overline{\psi}(y) \right] \left| 0 \right\rangle

where SF(x−y)S_F(x-y) is the Dirac propagator, ψ(x)\psi(x) and ψ‾(y)\overline{\psi}(y) are the Dirac fields at points xx and yy, respectively, and ∣0⟩\left| 0 \right\rangle is the vacuum state.

Derivation of Dirac Propagator in Peskin & Schroeder's Book

In Peskin and Schroeder's book, the Dirac propagator is derived by considering the commutator of two Dirac fields:

[ψa(x),ψ‾b(y)]=⟨0∣T[ψa(x)ψ‾b(y)]∣0⟩[\psi_a(x), \overline{\psi}_b(y)] = \left\langle 0 \right| T\left[ \psi_a(x) \overline{\psi}_b(y) \right] \left| 0 \right\rangle

Using the anticommutation relations for the Dirac fields, Peskin and Schroeder derive the following expression for the commutator:

[ψa(x),ψb(y)]={SF(x−y)\mboxifa=b−SF(x−y)\mboxifa≠b[\psi_a(x), \over{\psi}_b(y)] = \left\{ \begin{array}{ll} S_F(x-y) & \mbox{if } a=b \\ -S_F(x-y) & \mbox{if } a \neq b \end{array} \right.

where SF(x−y)S_F(x-y) is the Dirac propagator.

Properties of Dirac Propagator

The Dirac propagator has several important properties that are useful in QFT. Some of these properties include:

  • Translation invariance: The Dirac propagator is translationally invariant, meaning that it depends only on the difference between the coordinates xx and yy.
  • Causality: The Dirac propagator is causal, meaning that it vanishes for spacelike separations between xx and yy.
  • Hermiticity: The Dirac propagator is Hermitian, meaning that it satisfies the condition:

SF(x−y)=SF†(y−x)S_F(x-y) = S_F^\dagger(y-x)

Applications of Dirac Propagator

The Dirac propagator has numerous applications in QFT, including:

  • Fermion propagator: The Dirac propagator is used to calculate the propagator of fermions in QFT.
  • Fermion self-energy: The Dirac propagator is used to calculate the self-energy of fermions in QFT.
  • Fermion decay rates: The Dirac propagator is used to calculate the decay rates of fermions in QFT.

Conclusion

In conclusion, the Dirac propagator is a fundamental concept in QFT that describes the behavior of fermions in the presence of an external field. It is derived in Peskin and Schroeder's book by considering the commutator of two Dirac fields. The Dirac propagator has several important properties, including translation invariance, causality, and Hermiticity. It has numerous applications in QFT, including the calculation of fermion propagators, self-energies, and decay rates.

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.

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Q: What is the Dirac propagator?

A: The Dirac propagator is a fundamental concept in Quantum Field Theory (QFT) that describes the behavior of fermions in the presence of an external field. It is defined as the expectation value of the time-ordered product of two Dirac fields.

Q: How is the Dirac propagator derived in Peskin & Schroeder's book?

A: The Dirac propagator is derived in Peskin and Schroeder's book by considering the commutator of two Dirac fields. Using the anticommutation relations for the Dirac fields, they derive the following expression for the commutator:

[ψa(x),ψb(y)]={SF(x−y)\mboxifa=b−SF(x−y)\mboxifa≠b[\psi_a(x), \over{\psi}_b(y)] = \left\{ \begin{array}{ll} S_F(x-y) & \mbox{if } a=b \\ -S_F(x-y) & \mbox{if } a \neq b \end{array} \right.

where SF(x−y)S_F(x-y) is the Dirac propagator.

Q: What are the properties of the Dirac propagator?

A: The Dirac propagator has several important properties, including:

  • Translation invariance: The Dirac propagator is translationally invariant, meaning that it depends only on the difference between the coordinates xx and yy.
  • Causality: The Dirac propagator is causal, meaning that it vanishes for spacelike separations between xx and yy.
  • Hermiticity: The Dirac propagator is Hermitian, meaning that it satisfies the condition:

SF(x−y)=SF†(y−x)S_F(x-y) = S_F^\dagger(y-x)

Q: What are the applications of the Dirac propagator?

A: The Dirac propagator has numerous applications in QFT, including:

  • Fermion propagator: The Dirac propagator is used to calculate the propagator of fermions in QFT.
  • Fermion self-energy: The Dirac propagator is used to calculate the self-energy of fermions in QFT.
  • Fermion decay rates: The Dirac propagator is used to calculate the decay rates of fermions in QFT.

Q: What is the significance of the Dirac propagator in QFT?

A: The Dirac propagator is a fundamental concept in QFT that describes the behavior of fermions in the presence of an external field. It is a crucial tool for calculating various physical quantities, such as fermion propagators, self-energies, and decay rates.

Q: How does the Dirac propagator relate to the Dirac equation?

A: The Dirac propagator is closely related to the Dirac equation, which is a relativistic wave equation that describes the behavior of fermions. The Dirac propagator is used to calculate the propagator of fermions in QFT, which is a fundamental concept in the study of QFT.

Q: What are some common mistakes to avoid when working with the Dirac propagator?

A: Some common mistakes to avoid when working with the Dirac propagator include:

  • Failing to for the anticommutation relations: The Dirac propagator is derived using the anticommutation relations for the Dirac fields. Failing to account for these relations can lead to incorrect results.
  • Ignoring the properties of the Dirac propagator: The Dirac propagator has several important properties, including translation invariance, causality, and Hermiticity. Ignoring these properties can lead to incorrect results.

Q: What resources are available for learning more about the Dirac propagator?

A: There are several resources available for learning more about the Dirac propagator, including:

  • Peskin and Schroeder's book: Peskin and Schroeder's book is a comprehensive introduction to QFT, including the Dirac propagator.
  • Online lectures and tutorials: There are several online lectures and tutorials available that cover the Dirac propagator in detail.
  • Research papers: There are many research papers available that discuss the Dirac propagator in detail.

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.