Physlib.QFT.PerturbationTheory.WickAlgebra.TimeContraction

For a given field specification $\mathcal{F}$ and two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$, the time contraction $\text{timeContract}(\phi, \psi)$ is an element of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ defined as the difference between the time-ordered product and the normal-ordered product of the operators: $$\text{timeContract}(\phi, \psi) = \mathcal{T}(\phi\psi) - \mathcal{N}(\phi\psi)$$ where $\phi\psi$ denotes the product of the corresponding elements in the Wick algebra, $\mathcal{T}$ is the time-ordering operator, and $\mathcal{N}$ is the normal-ordering operator.

For a given field specification $\mathcal{F}$ and two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$, let $a = \text{ofFieldOp}(\phi)$ and $b = \text{ofFieldOp}(\psi)$ be their corresponding elements in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$. The time contraction of $\phi$ and $\psi$ is given by $$\text{timeContract}(\phi, \psi) = \mathcal{T}(a \cdot b) + (-1) \cdot \mathcal{N}(a \cdot b),$$ where $\cdot$ denotes the multiplication in the Wick algebra, $\mathcal{T}$ is the time-ordering operator, $\mathcal{N}$ is the normal-ordering operator, and $(-1) \cdot$ denotes scalar multiplication by $-1 \in \mathbb{C}$.

For a given field specification $\mathcal{F}$ and two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$, if $\phi$ and $\psi$ satisfy the time-ordering relation $\text{timeOrderRel}(\phi, \psi)$ (meaning $\phi$ is chronologically later than or equal to $\psi$), then the time contraction $\text{timeContract}(\phi, \psi)$ in the Wick algebra is equal to the super-commutator of the annihilation part of $\phi$ and the field operator $\psi$: $$\text{timeContract}(\phi, \psi) = [\text{anPart}(\phi), \text{ofFieldOp}(\psi)]_s$$ where $[\cdot, \cdot]_s$ denotes the super-commutator, $\text{anPart}(\phi)$ is the annihilation component of $\phi$, and $\text{ofFieldOp}(\psi)$ is the representation of $\psi$ in the Wick algebra.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$, if $\phi$ is not chronologically later than or equal to $\psi$ (i.e., the relation $\text{timeOrderRel}(\phi, \psi)$ does not hold), then the time contraction $\text{timeContract}(\phi, \psi)$ satisfies the following symmetry relation: $$\text{timeContract}(\phi, \psi) = \mathcal{S}(s(\phi), s(\psi)) \cdot \text{timeContract}(\psi, \phi)$$ where $s(\phi)$ and $s(\psi)$ are the field statistics (bosonic or fermionic) of the operators, and $\mathcal{S}$ is the exchange sign, which equals $-1$ if both operators are fermionic and $1$ otherwise.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$, if $\phi$ is not chronologically later than or equal to $\psi$ (i.e., $\neg \text{timeOrderRel}(\phi, \psi)$ holds), then the time contraction $\text{timeContract}(\phi, \psi)$ is given by: $$\text{timeContract}(\phi, \psi) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \psi) \cdot [\text{anPart}(\psi), \text{ofFieldOp}(\phi)]_s$$ where $\mathcal{F} \triangleright_s \phi$ and $\mathcal{F} \triangleright_s \psi$ are the field statistics of the operators, $\mathcal{S}$ is the exchange sign, $\text{anPart}(\psi)$ is the annihilation part of $\psi$, and $[\cdot, \cdot]_s$ denotes the super-commutator in the Wick algebra.

For a given field specification $\mathcal{F}$ and any two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$, the time contraction $\text{timeContract}(\phi, \psi)$ is determined by the chronological ordering of the operators as follows: - If $\text{timeOrderRel}(\phi, \psi)$ holds (i.e., $\phi$ is chronologically later than or equal to $\psi$), then: $$\text{timeContract}(\phi, \psi) = [\text{anPart}(\phi), \text{ofFieldOp}(\psi)]_s$$ - Otherwise, if $\neg \text{timeOrderRel}(\phi, \psi)$ holds, then: $$\text{timeContract}(\phi, \psi) = \mathcal{S}(\mathcal{F} \triangleright_s \phi, \mathcal{F} \triangleright_s \psi) \cdot [\text{anPart}(\psi), \text{ofFieldOp}(\phi)]_s$$ where $[\cdot, \cdot]_s$ denotes the super-commutator in the Wick algebra, $\text{anPart}$ is the annihilation part of the operator, $\mathcal{F} \triangleright_s \phi$ is the field statistic of $\phi$, and $\mathcal{S}$ is the exchange sign.

For a field specification $\mathcal{F}$ and any two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$, the time contraction $\text{timeContract}(\phi, \psi)$ is contained in the center of the Wick algebra $\mathcal{F}.\text{WickAlgebra}$ over the complex numbers $\mathbb{C}$.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$, if their field statistics are different (i.e., one is bosonic and the other is fermionic), then their time contraction vanishes: $$\text{timeContract}(\phi, \psi) = 0$$ where $\mathcal{F} \triangleright_s \phi$ denotes the statistic of the field operator.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \text{FieldOp}(\mathcal{F})$, the normal ordering of their time contraction vanishes: $$\mathcal{N}(\text{timeContract}(\phi, \psi)) = 0$$ where $\mathcal{N}$ denotes the normal ordering operator and $\text{timeContract}(\phi, \psi)$ is defined as the difference between the time-ordered product and the normal-ordered product of the operators, $\mathcal{T}(\phi\psi) - \mathcal{N}(\phi\psi)$.

Let $\mathcal{F}$ be a field specification. Let $\phi, \psi \in \mathcal{F}.\text{FieldOp}$ be two field operators that are contemporary, meaning both $\text{timeOrderRel}(\phi, \psi)$ and $\text{timeOrderRel}(\psi, \phi)$ hold (typically implying they exist at the same time). For any elements $a$ and $b$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the time-ordering operator $\mathcal{T}$ satisfies: $$\mathcal{T}(a \cdot \text{timeContract}(\phi, \psi) \cdot b) = \text{timeContract}(\phi, \psi) \cdot \mathcal{T}(a \cdot b)$$ where $\text{timeContract}(\phi, \psi)$ is the time contraction of the two operators.

Let $\mathcal{F}$ be a field specification. Let $\phi, \psi \in \mathcal{F}.\text{FieldOp}$ be two field operators that are contemporary, meaning both $\text{timeOrderRel}(\phi, \psi)$ and $\text{timeOrderRel}(\psi, \phi)$ hold. For any element $b$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the time-ordering operator $\mathcal{T}$ satisfies: $$\mathcal{T}(\text{timeContract}(\phi, \psi) \cdot b) = \text{timeContract}(\phi, \psi) \cdot \mathcal{T}(b)$$ where $\text{timeContract}(\phi, \psi)$ denotes the time contraction of the two operators.

Let $\mathcal{F}$ be a field specification. For any two field operators $\phi, \psi \in \mathcal{F}.\text{FieldOp}$ that are not contemporary (i.e., the condition $\text{timeOrderRel}(\phi, \psi) \wedge \text{timeOrderRel}(\psi, \phi)$ does not hold), the time-ordering operator $\mathcal{T}$ applied to their time contraction in the Wick algebra is zero: $$\mathcal{T}(\text{timeContract}(\phi, \psi)) = 0$$ where $\text{timeContract}(\phi, \psi)$ is the difference between the time-ordered and normal-ordered products of the operators.

Let $\mathcal{F}$ be a field specification. For any two incoming asymptotic field operators $\phi_{\text{in}}$ and $\psi_{\text{in}}$ in the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, their time contraction is zero: $$\text{timeContract}(\phi_{\text{in}}, \psi_{\text{in}}) = 0$$ This implies that in the context of Wick's theorem, there are no contractions between two incoming asymptotic fields, effectively preventing Feynman diagrams where two incoming vertices are connected to each other.

For a given field specification $\mathcal{F}$, let $\phi_{\text{out}}(\varphi)$ and $\phi_{\text{out}}(\psi)$ be two outgoing asymptotic field operators, where $\varphi$ and $\psi$ each represent a configuration consisting of a field, an asymptotic label, and a 3-momentum. In the Wick algebra $\mathcal{F}.\text{WickAlgebra}$, the time contraction of these two outgoing asymptotic operators is zero: $$\text{timeContract}(\phi_{\text{out}}(\varphi), \phi_{\text{out}}(\psi)) = 0$$ This implies that in the context of Wick's theorem, there are no contractions between two outgoing asymptotic fields, effectively preventing Feynman diagrams where outgoing vertices are connected directly to other outgoing vertices.