Lagrangian and Hamiltonian Formalism for Fields

Quantum Field Theory (QFT) provides a powerful framework for describing elementary particles and their interactions. At its core lies the concept of fields, and understanding how these fields evolve and interact is crucial. The Lagrangian and Hamiltonian formalisms, borrowed from classical mechanics, are fundamental tools for this purpose.

The Lagrangian Formalism for Fields

In classical mechanics, the Lagrangian is defined as the difference between kinetic and potential energy ( $L = T - V$ ). For fields, we generalize this to a Lagrangian density, $\mathcal{L}$ , which is a function of the field(s) and their derivatives. The action, $S$ , is the integral of the Lagrangian density over spacetime: $S = \int d^4x \,\mathcal{L}(\phi, \partial_\mu \phi)$ . The principle of least action dictates that the physical evolution of the field is such that the action is stationary.

The Euler-Lagrange equations are the equations of motion for fields.

Applying the principle of least action to the action integral leads to the Euler-Lagrange equations for fields. These equations are fundamental for deriving the dynamics of any given field.

The variation of the action, $\delta S$ , must be zero for the physical path. This leads to the Euler-Lagrange equations for fields: $\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) = 0$ . These are the field-theoretic analogues of Newton's second law.

What is the fundamental quantity integrated over spacetime to define the action in field theory?

The Lagrangian density.

The Hamiltonian Formalism for Fields

The Hamiltonian formalism offers an alternative perspective, focusing on the canonical momentum and the Hamiltonian. The canonical momentum conjugate to a field $\phi$ is defined as $\pi = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}$ . The Hamiltonian density, $\mathcal{H}$ , is obtained via a Legendre transformation of the Lagrangian density: $\mathcal{H} = \pi \frac{\partial \phi}{\partial t} - \mathcal{L}$ . The total Hamiltonian is the integral of the Hamiltonian density over space: $H = \int d^3x \,\mathcal{H}$ .

Hamilton's equations describe the time evolution of fields and their conjugate momenta.

Similar to classical mechanics, Hamilton's equations in field theory relate the time derivatives of fields and their conjugate momenta to the Hamiltonian density.

The time evolution of the field and its conjugate momentum are governed by Hamilton's equations: $\frac{\partial \phi}{\partial t} = \frac{\delta H}{\delta \pi}$ and $\frac{\partial \pi}{\partial t} = -\frac{\delta H}{\delta \phi}$ . Here, $\frac{\delta H}{\delta \phi}$ represents a functional derivative.

Feature	Lagrangian Formalism	Hamiltonian Formalism
Primary Quantity	Lagrangian Density ( $\mathcal{L}$ )	Hamiltonian Density ( $\mathcal{H}$ )
Key Variables	Field ( $\phi$ ), Field Derivatives ( $\partial_\mu \phi$ )	Field ( $\phi$ ), Conjugate Momentum ( $\pi$ )
Equations of Motion	Euler-Lagrange Equations	Hamilton's Equations
Transformation	Directly from physical principles	Legendre Transform of Lagrangian Density

The transition from Lagrangian to Hamiltonian formalism involves a Legendre transformation. Imagine a point on a curve (Lagrangian) defined by its position and slope. The Legendre transform maps this point to a new point defined by the slope and the intercept of the tangent line at that point. In field theory, the 'curve' is the Lagrangian density as a function of field derivatives, and the 'slope' is the conjugate momentum. The transformation allows us to switch from describing the system using fields and their velocities (derivatives) to using fields and their momenta.

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Symmetries and Conservation Laws (Noether's Theorem)

A profound connection exists between symmetries of the Lagrangian and conserved quantities, as described by Noether's theorem. For every continuous symmetry of the action, there exists a corresponding conserved current and a conserved charge. This theorem is a cornerstone of QFT, linking fundamental symmetries (like Lorentz invariance or gauge invariance) to conserved quantities (like energy, momentum, angular momentum, and electric charge).

Noether's theorem is a powerful tool that reveals the deep relationship between the symmetries of a physical system and the quantities that remain constant over time.

Applications in Quantum Field Theory

The Lagrangian and Hamiltonian formalisms are indispensable for constructing quantum field theories. For instance, the Klein-Gordon equation for scalar fields and the Dirac equation for fermionic fields are derived from specific Lagrangian densities. Quantization procedures, such as canonical quantization, directly utilize the Hamiltonian formulation to define commutation relations and construct the quantum operators for fields and their conjugate momenta.

What theorem connects continuous symmetries of the Lagrangian to conserved quantities?

Noether's theorem.

Learning Resources

Quantum Field Theory by Mark Srednicki - Chapter 1: The Lagrangian and Hamiltonian Formalisms(documentation)

This is the first chapter of a widely respected QFT textbook, providing a rigorous introduction to the Lagrangian and Hamiltonian formalisms for fields.

Introduction to Quantum Field Theory - Lecture Notes by David Tong(documentation)

David Tong's comprehensive lecture notes cover the foundational aspects of QFT, including detailed explanations of Lagrangian and Hamiltonian mechanics for fields.

Classical Field Theory - Lecture Notes by John Preskill(documentation)

These notes provide a thorough grounding in classical field theory, which is essential before diving into quantum aspects, with clear explanations of the formalisms.

Lagrangian and Hamiltonian Mechanics - Wikipedia(wikipedia)

A foundational overview of the classical mechanics concepts that are extended to field theory, useful for refreshing core principles.

Noether's Theorem - Wikipedia(wikipedia)

Explains Noether's theorem, its significance in physics, and its application to symmetries and conservation laws in classical and quantum field theory.

Quantum Field Theory for the Gifted Amateur - Chapter 2: Lagrangians(book_excerpt)

While a book, this resource often has accessible online summaries or introductory chapters that clearly explain the Lagrangian approach to fields.

Introduction to Quantum Field Theory - Lecture 3: Lagrangians and Hamiltonians(video)

A video lecture that breaks down the Lagrangian and Hamiltonian formalisms in the context of quantum field theory, offering a visual and auditory explanation.

Covariant Canonical Quantization - A Modern Approach(paper)

This paper delves into more advanced aspects of quantization, often referencing the foundational Lagrangian and Hamiltonian frameworks for fields.

The Path Integral Formulation of Quantum Mechanics(documentation)

While focused on path integrals, this resource often contextualizes them within the Lagrangian framework, highlighting its importance in QFT.

Feynman Lectures on Physics Vol. III - Chapter 14: The Relation of Wave and Matrix Mechanics(documentation)

Feynman's classic lectures touch upon the underlying principles that bridge classical mechanics, including Lagrangian and Hamiltonian ideas, to quantum mechanics.