3.1 Free scalar field theory and the Klein-Gordon equation

Free scalar field theory introduces us to the Klein-Gordon equation, a fundamental concept in quantum field theory. This equation describes the behavior of scalar fields, which are the simplest type of quantum fields. It's like the gateway to understanding more complex fields.

The Klein-Gordon equation connects classical field theory with quantum mechanics. It shows how particles can be viewed as excitations of fields, laying the groundwork for understanding particle interactions in quantum field theory. This is crucial for grasping the rest of the chapter.

Klein-Gordon Equation for Scalar Fields

Lagrangian Density and Action for Scalar Fields

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• The Lagrangian density for a free scalar field is given by $L = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m^2\phi^2$
  • $\phi$ represents the scalar field
  • $\partial_\mu$ denotes the partial derivative with respect to the spacetime coordinate $x^\mu$
  • $m$ is the mass of the scalar field
• The action for the scalar field is the integral of the Lagrangian density over spacetime $S = \int d^4x L$
  • The action principle states that the physical dynamics of a system is determined by the stationary points of the action

Deriving the Klein-Gordon Equation

• The principle of least action states that the variation of the action with respect to the field should vanish $\frac{\delta S}{\delta\phi} = 0$
  • This leads to the Euler-Lagrange equation for the scalar field
• Applying the Euler-Lagrange equation to the Lagrangian density of the free scalar field yields the Klein-Gordon equation $(\partial_\mu\partial^\mu + m^2)\phi = 0$
  • This is a second-order partial differential equation that describes the dynamics of the scalar field
  • The Klein-Gordon equation is Lorentz invariant, ensuring that the scalar field behaves consistently in all inertial reference frames

Physical Meaning of Klein-Gordon Equation

Relativistic Wave Equation for Scalar Fields

• The Klein-Gordon equation is a relativistic wave equation that describes the dynamics of a free scalar field
  • It is the quantum field theory analog of the Schrödinger equation for a free particle in non-relativistic quantum mechanics
  • The Klein-Gordon equation incorporates both the wave-like and particle-like properties of the scalar field
• The Klein-Gordon equation is second-order in time, which implies that the scalar field has two independent solutions
  • These solutions correspond to positive and negative energy states
  • The presence of negative energy states is a consequence of the relativistic nature of the equation

Particles and Antiparticles

• The presence of negative energy states in the Klein-Gordon equation leads to the interpretation of the scalar field as a collection of particles and antiparticles
  • Particles are associated with positive energy states, while antiparticles are associated with negative energy states
  • The creation of a particle is accompanied by the annihilation of an antiparticle, and vice versa, ensuring the conservation of energy
• The mass term $m^2$ in the Klein-Gordon equation determines the rest mass of the scalar field quanta (particles) associated with the field
  • The mass term provides a lower bound on the energy of the particles
  • Massless scalar fields, such as the Goldstone boson, have $m = 0$ and exhibit different behavior compared to massive fields

Plane Wave Solutions of Klein-Gordon Equation

Superposition Principle and General Solution

• The Klein-Gordon equation is a linear partial differential equation, so the superposition principle applies
  • The general solution can be written as a linear combination of plane wave solutions
  • This allows for the construction of more complex field configurations by superposing plane waves
• Plane wave solutions to the Klein-Gordon equation have the form $\phi(x) = A \exp(\pm ik_\mu x^\mu)$
  • $A$ is a complex amplitude
  • $k_\mu$ is the four-momentum
  • $x^\mu$ is the spacetime coordinate

Dispersion Relation and Energy States

• The four-momentum $k_\mu$ satisfies the relativistic dispersion relation $k_\mu k^\mu = m^2$
  • This relation connects the energy and momentum of the scalar field quanta
  • It ensures that the scalar field quanta obey the relativistic energy-momentum relation $E^2 = \vec{p}^2 + m^2$
• The positive and negative frequency solutions correspond to the positive and negative energy states, respectively
  • Positive frequency solutions describe particles, while negative frequency solutions describe antiparticles
  • The choice of positive or negative frequency determines the particle or antiparticle nature of the field quanta

Quantization of Scalar Fields

Canonical Quantization Procedure

• Quantization of the scalar field is achieved by promoting the field and its conjugate momentum to operators satisfying the canonical commutation relations $[\phi(x), \pi(y)] = i\delta^3(\vec{x} - \vec{y})$
  • $\pi(x) = \partial_0\phi(x)$ is the conjugate momentum
  • The commutation relations ensure that the field operators obey the uncertainty principle
• The scalar field operator can be expanded in terms of plane wave solutions and creation and annihilation operators $\phi(x) = \int \frac{d^3k}{(2\pi)^3} \frac{1}{\sqrt{2\omega_k}} \left(a_k \exp(-ik_\mu x^\mu) + a_k^\dagger \exp(ik_\mu x^\mu)\right)$
  • $a_k$ and $a_k^\dagger$ are the annihilation and creation operators, respectively
  • $\omega_k = \sqrt{\vec{k}^2 + m^2}$ is the energy of the mode with momentum $\vec{k}$

Creation and Annihilation Operators

• The creation operator $a_k^\dagger$ creates a scalar field quantum (particle) with momentum $\vec{k}$
  • Acting on the vacuum state $|0\rangle$ with the creation operator produces a one-particle state $|1_k\rangle = a_k^\dagger|0\rangle$
  • Multiple applications of creation operators generate multi-particle states
• The annihilation operator $a_k$ destroys a scalar field quantum with momentum $\vec{k}$
  • Acting on a one-particle state with the annihilation operator returns the vacuum state $a_k|1_k\rangle = |0\rangle$
  • The annihilation operator acting on the vacuum state gives zero $a_k|0\rangle = 0$
• The creation and annihilation operators satisfy the commutation relations $[a_k, a_{k'}^\dagger] = \delta^3(\vec{k} - \vec{k}'), \quad [a_k, a_{k'}] = [a_k^\dagger, a_{k'}^\dagger] = 0$
  • These commutation relations ensure the bosonic nature of the scalar field quanta
  • They also imply that the creation and annihilation operators for different momenta commute with each other

Vacuum State and Multi-Particle States

• The vacuum state $|0\rangle$ is defined as the state annihilated by all annihilation operators $a_k|0\rangle = 0 \quad \text{for all } \vec{k}$
  • The vacuum state represents the absence of any scalar field quanta
  • It is the lowest energy state of the quantum scalar field
• Multi-particle states are constructed by applying creation operators to the vacuum state
  • A two-particle state with momenta $\vec{k}_1$ and $\vec{k}_2$ is given by $|1_{k_1}, 1_{k_2}\rangle = a_{k_1}^\dagger a_{k_2}^\dagger|0\rangle$
  • The number of particles in a state is determined by the number of creation operators applied to the vacuum
• The scalar field quantum states form a complete basis for the Hilbert space of the quantum scalar field theory
  • Any state in the Hilbert space can be expressed as a linear combination of the multi-particle states
  • The scalar field operators and the creation/annihilation operators allow for the computation of observables and the study of the dynamics of the quantum scalar field