8.2 Functional integral for scalar field theory

Quantum field theory takes the path integral concept to a whole new level. Instead of particle paths, we're now dealing with fields that span all of spacetime. It's like upgrading from a single-player game to a massive multiplayer universe!

The functional integral is the star of the show here. It lets us calculate important stuff like transition amplitudes and correlation functions by considering all possible field configurations. Think of it as a cosmic suggestion box where every field gets a vote.

Path Integral Formalism for Quantum Fields

Generalizing Path Integrals to Quantum Field Theory

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• The path integral formalism, originally developed for quantum mechanics, can be generalized to quantum field theory by replacing the finite-dimensional path integral with an infinite-dimensional functional integral
• In quantum field theory, the dynamical variables are fields $\phi(x)$ defined at each point in spacetime, rather than particle positions $q(t)$ evolving in time
  • Fields are functions that assign a value to each point in spacetime (scalar fields, vector fields, tensor fields)
  • Example: The electromagnetic field is described by a vector field $A_\mu(x)$
• The transition amplitude between initial and final field configurations is given by a functional integral over all possible field configurations, weighted by the exponential of the action
  • The functional integral sums over all possible field configurations connecting the initial and final states
  • The action $S[\phi]$ is a functional that assigns a number to each field configuration, determining its weight in the integral

Functional Integral in Scalar Field Theory

• For a scalar field theory, the functional integral is written as $Z = \int \mathcal{D}\phi \exp(iS[\phi])$, where $\mathcal{D}\phi$ denotes the integration measure over all possible field configurations $\phi(x)$
  • The integration measure $\mathcal{D}\phi$ is a product of infinitesimal integrals over the field values at each spacetime point
  • Example: For a discrete spacetime lattice, $\mathcal{D}\phi = \prod_x d\phi(x)$
• The functional integral can be used to calculate vacuum-to-vacuum transition amplitudes, as well as correlation functions of the scalar field
  • Vacuum-to-vacuum transition amplitude: $\langle 0_\text{out} | 0_\text{in} \rangle = \int \mathcal{D}\phi \exp(iS[\phi])$
  • Correlation functions: $\langle \phi(x_1) \ldots \phi(x_n) \rangle = \frac{1}{Z} \int \mathcal{D}\phi \phi(x_1) \ldots \phi(x_n) \exp(iS[\phi])$

Functional Integral for Scalar Fields

Lagrangian Density and Action for Scalar Fields

• The action $S[\phi]$ is a functional of the scalar field $\phi(x)$ and is given by the integral of the Lagrangian density $\mathcal{L}$ over spacetime: $S[\phi] = \int d^n x \mathcal{L}(\phi, \partial_\mu\phi)$
  • The Lagrangian density is a function of the field and its derivatives at each spacetime point
  • The action is a scalar quantity that is invariant under Lorentz transformations
• The Lagrangian density for a free scalar field theory is $\mathcal{L} = \frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi) - \frac{1}{2}m^2\phi^2$, where $m$ is the mass of the scalar field
  • The first term represents the kinetic energy of the field, involving the spacetime derivatives of the field
  • The second term represents the potential energy, which is quadratic in the field for a free theory
  • Example: For a massless scalar field ($m=0$), the Lagrangian density is $\mathcal{L} = \frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)$

Evaluating the Functional Integral

• The functional integral $Z = \int \mathcal{D}\phi \exp(iS[\phi])$ can be evaluated using various techniques, depending on the complexity of the theory
• For free field theories, the functional integral is Gaussian and can be evaluated exactly using functional methods
  • The result is expressed in terms of the Green's functions or propagators of the theory
  • Example: For a free scalar field, the propagator is $\langle \phi(x) \phi(y) \rangle = \int \frac{d^n k}{(2\pi)^n} \frac{e^{-ik(x-y)}}{k^2 - m^2 + i\epsilon}$
• For interacting field theories, perturbative techniques such as Feynman diagrams can be used to evaluate the functional integral order by order in the coupling constant
  • The interacting theory is treated as a perturbation around the free theory
  • Feynman diagrams represent the various ways in which fields can interact and propagate
  • Example: For a $\phi^4$ interaction term $\mathcal{L}_\text{int} = -\frac{\lambda}{4!}\phi^4$, the lowest-order Feynman diagram is the four-point vertex

Action and Lagrangian in Functional Integral

Role of the Action in the Functional Integral

• The action $S[\phi]$ plays a central role in the functional integral formalism, as it determines the weight assigned to each field configuration in the integral
  • Field configurations with a smaller action are more likely to contribute to the integral
  • The classical field configuration, which minimizes the action, dominates the functional integral in the classical limit ($\hbar \to 0$)
• The principle of stationary action, $\delta S[\phi] = 0$, leads to the classical field equations of motion
  • Varying the action with respect to the field yields the Euler-Lagrange equations
  • Example: For a scalar field, the Euler-Lagrange equation is $\partial_\mu(\partial^\mu\phi) + m^2\phi = 0$ (Klein-Gordon equation)

Lagrangian Density and Field Dynamics

• The Lagrangian density $\mathcal{L}(\phi, \partial_\mu\phi)$ encodes the dynamics of the scalar field theory, including the kinetic and potential energy terms
  • The kinetic term determines how the field propagates through spacetime
  • The potential term determines the interactions between the field and itself or other fields
• The form of the Lagrangian density is constrained by symmetry principles and the requirement of renormalizability
  • Symmetries of the Lagrangian lead to conserved quantities (Noether's theorem)
  • Example: The U(1) symmetry of the complex scalar field Lagrangian leads to the conservation of charge
• The Lagrangian density can be modified to include interactions between fields, such as the $\phi^4$ interaction term $\mathcal{L}_\text{int} = -\frac{\lambda}{4!}\phi^4$
  • Interaction terms introduce non-linearities in the field equations and enable the description of scattering processes
  • The coupling constant $\lambda$ determines the strength of the interaction

Correlation Functions from Functional Integral

Definition and Properties of Correlation Functions

• Correlation functions, also known as Green's functions, are expectation values of products of field operators at different spacetime points
  • They describe the correlations between field values at different locations and times
  • The n-point correlation function is given by $\langle \phi(x_1) \ldots \phi(x_n) \rangle = \frac{1}{Z} \int \mathcal{D}\phi \phi(x_1) \ldots \phi(x_n) \exp(iS[\phi])$, where $Z$ is the partition function
• Correlation functions satisfy various properties and symmetries, such as translational invariance, Lorentz invariance, and causality
  • Translational invariance: $\langle \phi(x_1+a) \ldots \phi(x_n+a) \rangle = \langle \phi(x_1) \ldots \phi(x_n) \rangle$
  • Lorentz invariance: The correlation functions are invariant under Lorentz transformations of the spacetime coordinates
  • Causality: The correlation functions vanish for spacelike separated points (outside the light cone)

Propagators and Feynman Diagrams

• The two-point correlation function, or propagator, describes the probability amplitude for a particle to propagate from one spacetime point to another
  • It is the inverse of the differential operator appearing in the quadratic part of the action
  • Example: For a scalar field, the propagator satisfies $(\partial_\mu\partial^\mu + m^2) \langle \phi(x) \phi(y) \rangle = -i\delta^{(n)}(x-y)$
• Higher-order correlation functions contain information about the interactions between fields and can be used to calculate scattering amplitudes
  • They are related to the S-matrix elements, which describe the transition probabilities between initial and final states
  • Example: The four-point correlation function $\langle \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \rangle$ is related to the scattering amplitude for $2 \to 2$ particle scattering
• Perturbative techniques, such as Feynman diagrams, can be used to evaluate correlation functions in interacting field theories
  • Feynman diagrams represent the various ways in which particles can interact and propagate
  • Each diagram corresponds to a term in the perturbative expansion of the correlation function
  • The rules for constructing and evaluating Feynman diagrams are derived from the functional integral formalism