Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part I, Chapter 1 | Page 3 of 8

Hamiltonian Formalism for Fields

Canonical momentum and the Hamiltonian density

1.7 Canonical Momentum for Fields

Just as in particle mechanics, we can perform a Legendre transform to obtain the Hamiltonian formulation.

Conjugate Momentum Density

The canonical momentum density (or conjugate momentum) is defined as:

$$\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}(x)} = \frac{\partial \mathcal{L}}{\partial(\partial_0 \phi)}$$

where $\dot{\phi} = \partial_0 \phi = \partial \phi/\partial t$. Note that Ο€ is a density in space, not a 4-vector.

Example: Real Scalar Field

For the Klein-Gordon Lagrangian:

$$\mathcal{L} = \frac{1}{2}(\partial_t \phi)^2 - \frac{1}{2}(\nabla \phi)^2 - \frac{1}{2}m^2 \phi^2$$

The canonical momentum is:

$$\pi(x) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = \dot{\phi}(x)$$

So for the Klein-Gordon field, the canonical momentum equals the time derivative of the field.

1.8 Hamiltonian Density

The Hamiltonian density is obtained via Legendre transform:

$$\mathcal{H}(x) = \pi(x) \dot{\phi}(x) - \mathcal{L}(x)$$

The Hamiltonian (total energy) is:

$$H = \int d^3x \, \mathcal{H}(x)$$

Example: Klein-Gordon Field

For the real scalar field with Ο€ = $\dot{\phi}$:

$$\mathcal{H} = \pi \dot{\phi} - \mathcal{L} = \pi^2 - \left[\frac{1}{2}\pi^2 - \frac{1}{2}(\nabla \phi)^2 - \frac{1}{2}m^2 \phi^2\right]$$

Simplifying:

$$\boxed{\mathcal{H} = \frac{1}{2}\pi^2 + \frac{1}{2}(\nabla \phi)^2 + \frac{1}{2}m^2 \phi^2}$$

This has the familiar structure: kinetic + gradient + potential energy.

The total Hamiltonian:

$$H = \int d^3x \left[\frac{1}{2}\pi^2 + \frac{1}{2}(\nabla \phi)^2 + \frac{1}{2}m^2 \phi^2\right]$$

This is positive definite (H β‰₯ 0), as required for stability.

1.9 Hamilton's Equations for Fields

The field Ο†(x) and its canonical momentum Ο€(x) are treated as independent dynamical variables. Hamilton's equations are:

$$\dot{\phi}(x) = \frac{\delta H}{\delta \pi(x)}, \quad \dot{\pi}(x) = -\frac{\delta H}{\delta \phi(x)}$$

where $\delta/\delta \phi(x)$ denotes the functional derivative:

$$\frac{\delta F[\phi]}{\delta \phi(x)} = \lim_{\epsilon \to 0} \frac{F[\phi + \epsilon \delta^3(x - y)] - F[\phi]}{\epsilon}$$

Derivation for Klein-Gordon

The Hamiltonian density is:

$$\mathcal{H} = \frac{1}{2}\pi^2 + \frac{1}{2}(\nabla \phi)^2 + \frac{1}{2}m^2 \phi^2$$

First equation:

$$\dot{\phi}(x) = \frac{\delta H}{\delta \pi(x)} = \frac{\delta}{\delta \pi(x)} \int d^3y \, \mathcal{H}(y) = \pi(x)$$

This recovers Ο€ = $\dot{\phi}$.

Second equation:

$$\dot{\pi}(x) = -\frac{\delta H}{\delta \phi(x)} = -\frac{\delta}{\delta \phi(x)} \int d^3y \left[\frac{1}{2}(\nabla_y \phi(y))^2 + \frac{1}{2}m^2 \phi(y)^2\right]$$

Using integration by parts on the gradient term:

$$\int d^3y \, (\nabla_y \phi) \cdot \nabla_y \left(\frac{\delta \phi(y)}{\delta \phi(x)}\right) = \int d^3y \, (\nabla_y \phi) \cdot \nabla_y \delta^3(y - x) = -\nabla^2 \phi(x)$$

Therefore:

$$\dot{\pi}(x) = \nabla^2 \phi(x) - m^2 \phi(x)$$

Combining the two Hamilton equations:

$$\ddot{\phi} = \nabla^2 \phi - m^2 \phi$$

which is the Klein-Gordon equation! Hamilton's equations are equivalent to the Euler-Lagrange equation.

1.10 Path to Quantization

The Hamiltonian formalism is essential for canonical quantization. The procedure:

  1. Classical theory: Ο†(x) and Ο€(x) are classical fields satisfying Hamilton's equations
  2. Quantization: Promote Ο† and Ο€ to operators $\hat{\phi}(x)$ and $\hat{\pi}(x)$
  3. Canonical commutation relations:
    $$[\hat{\phi}(\vec{x}, t), \hat{\pi}(\vec{y}, t)] = i\hbar \delta^3(\vec{x} - \vec{y})$$
  4. Hamiltonian operator: $\hat{H}$ generates time evolution

This procedure, called canonical quantization, is the foundation of quantum field theory. We'll develop it fully in Part II.

The Hamiltonian formalism makes the phase space structure (Ο†, Ο€) explicit, which is crucial for understanding the degrees of freedom and for quantization.

Key Concepts (Page 3)

  • β€’ Canonical momentum density: Ο€ = $\partial \mathcal{L}/\partial \dot{\phi}$
  • β€’ Hamiltonian density: $\mathcal{H} = \pi \dot{\phi} - \mathcal{L}$
  • β€’ Hamilton's equations equivalent to Euler-Lagrange
  • β€’ (Ο†, Ο€) form a phase space for the field
  • β€’ Hamiltonian formalism essential for canonical quantization
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