Free field - Citizendia

Classically, a free field has equations of motion given by linear partial differential equations. The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Such linear PDE's have a unique solution for a given initial condition.

In quantum field theory, an operator valued distribution is a free field if it satisfies some linear partial differential equations such that the corresponding case of the same linear PDEs for a classical field (i. In quantum field theory (QFT the forces between particles are mediated by other particles In Physics the Wightman axioms are an attempt at a mathematically rigorous formulation of Quantum field theory. The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i e. not an operator) would be the Euler-Lagrange equation for some quadratic Lagrangian. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions In mathematics the term quadratic describes something that pertains to squares, to the operation of Squaring, to terms of the second degree, or equations The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system We can differentiate distributions by defining their derivatives via differentiated test functions. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions See Schwartz distribution for more details. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions Since we are dealing not with ordinary distributions but operator valued distributions, it is understood these PDEs aren't constraints on states but instead a description of the relations among the smeared fields. Beside the PDEs, the operators also satisfy another relation, the commutation/anticommutation relations.

Basically, commutator (for bosons)/anticommutator (for fermions) of two smeared fields is i times the Peierls bracket of the field with itself (which is really a distribution, not a function) for the PDE's smeared over both test functions. In Theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. This has the form of a CCR/CAR algebra. In Quantum field theory, if V is a real Vector space equipped with a Nonsingular real Antisymmetric Bilinear form ( (i

CCR/CAR algebras with infinitely many degrees of freedom have many inequivalent irreducible unitary representations. If the theory is defined over Minkowski space, we may choose the unitary irrep containing a vacuum state although that isn't always necessary. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Quantum field theory, the vacuum state (also called the vacuum) is the Quantum state with the lowest possible Energy.

Example

Let φ be an operator valued distribution and the (Klein-Gordon) PDE be

$\partial^\mu \partial_\mu \phi+m^2 \phi^2=0$ .

This is a bosonic field. Let's call the distribution given by the Peierls bracket Δ.

Then,

{φ(x),φ(y)} = Δ(x; y)

where here, φ is a classical field and {,} is the Peierls bracket.

Then, the canonical commutation relation relation is

$[\phi[f],\phi[g]]=i\Delta[f,g] \,$ . In Physics, the canonical commutation relation is the relation between Canonical conjugate quantities (quantities which are related by definition such that one is

Note that Δ is a distribution over two arguments, and so, can be smeared as well.

Equivalently, we could have insisted that

$\mathcal{T}\{[((\partial^\mu \partial_\mu+m^2)\phi)[f],\phi[g]]\}=-i\int d^dx f(x)g(x)$

where $\mathcal{T}$ is the time ordering operator and that if the supports of f and g are spacelike separated,

[φ[f],φ[g]] = 0

. In Theoretical physics, path-ordering is the procedure (or a meta-operator {\mathcal P} of ordering a product of many operators according to the value of one