PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Quantum field theory → Second Quantization ↓


Path Integral in Second Quantization


Path integrals in second quantization provide a powerful framework for understanding quantum field theories. These integrals help physicists model the dynamics of quantum fields using a mathematical approach that differs from traditional wave functions. Instead of focusing on particle states, path integrals emphasize fields - continuous entities filling space, which can oscillate and interact.

Introduction to path integrals

At a high level, path integration is a way of handling quantum dynamical systems that integrates over all possible paths a system can take between two states. This method considers all possible histories of a system. Richard Feynman introduced this formalism, which beautifully connects quantum mechanics to classical mechanics.

What is path integral?

In its simplest form, consider a particle traveling from one point to another. The path integral is the sum of all possible trajectories the particle can follow from its starting point to its ending point. Each trajectory is weighed by an exponential factor associated with the classical action determined by that path:

      ∫ D[x(t)] E (i⁄ℏ)S[x(t)]

where D[x(t)] denotes the sum over all paths x(t), and S[x(t)] is the classical action.

An example in one dimension

Imagine a particle moving back and forth on a one-dimensional line between two points, a and b. The classical path is a straight line between these points. Path integrals ask us to consider not only this straight line path, but also any zigzag or irregular paths. Mathematically, this is expressed as:

      ∫ D[x(t)] E (i⁄ℏ)S[x(t)]

where each path contributes to the total, giving us the probability amplitude for the particle to travel from a to b.

Second quantization framework

Second quantization is an extension of quantum mechanics designed to handle many-body systems, such as electrons in an atom or molecules in a condensed matter system. The idea is to quantize fields rather than particles. This approach shifts the focus from particles, which are discrete entities, to fields that act continuously over space and time.

Why use second quantization?

Second quantization is particularly useful in scenarios involving indistinguishable particles, such as bosons or fermions. This formalism manages creation and annihilation operators, which represent the addition or removal of particles from certain states.

Territories as fundamental goods

In the second quantization, fields become the fundamental units. Quantum fields can create or annihilate particles via operators. For example, the creation operator adds a particle to a state, while the annihilation operator removes a particle.

Example: simple harmonic oscillator

The quantum field theory of the harmonic oscillator in the second quantization involves operators acting on fields where the states are described by quantized energy levels.

      |n⟩ = (a † ) n / √(n!) |0⟩

The creation operator a † moves the system to a higher energy level, while a, the annihilation operator, lowers the energy level.

Bridging path integration and second quantization

Path integrals in second quantization allow an elegant description of quantum fields. In simple terms consider a scalar field ϕ(x):

      S[ϕ] = ∫ d d x dt L(ϕ, ∂ μ ϕ)

where L is the Lagrangian density. The path integral is now over ϕ(x) rather than over the simple path x(t):

      ∫ D[ϕ(x)] e (i⁄ℏ)S[ϕ(x)]

Field path integral

The field path integral sums over all configurations of the field in spacetime. Instead of integrating over the paths of a particle, we integrate over all possible field configurations at every point.

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Applications in quantum field theory

Path integrals in second quantization provide important insights into complex systems in quantum field theory.

Interactive area

Path integrals can model interactions between fields, which is important in particle physics. Consider two interacting scalar fields, ϕ and χ:

      L(ϕ,χ) = 0.5(∂ μ ϕ)^2 + 0.5m 2 ϕ 2 + 0.5(∂ μ χ)^2 + 0.5M 2 χ 2 + gϕ 2 χ

The coupling term gϕ 2 χ represents how ϕ and χ fields interact, which is calculated via a path integral over the two fields.

Renormalization

In quantum field theory, we often encounter infinities that path integrals help manage through renormalization. Path integrals allow us to systematically sum over fluctuations in the field, providing a framework for handling these infinities.

Example: quantum electrodynamics (QED)

In QED, the path integral formalism allows us to calculate interactions involving electrons and photons. The renormalization process helps address infinities arising from loop diagrams that represent virtual particles in Feynman diagrams.

Conclusion

Path integrals in second quantization expand our ability to model complex quantum systems. By integrating over all possible field configurations, they provide deep insights into the behavior of quantum fields and interactions. This formalism is the cornerstone of modern quantum field theory, which underlies much of our understanding in areas such as particle physics and condensed matter systems.


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