Correlation function (quantum field theory)

For other uses, see Correlation function (disambiguation).

In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements.

Definition

For a scalar field theory with a single field ${\displaystyle \phi (x)}$  and a vacuum state ${\displaystyle |\Omega \rangle }$  at every event (x) in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of ${\displaystyle n}$  field operators in the Heisenberg picture $${\displaystyle G_{n}(x_{1},\dots ,x_{n})=\langle \Omega |T\{{\mathcal {\phi }}(x_{1})\dots {\mathcal {\phi }}(x_{n})\}|\Omega \rangle .}$$ 

Here ${\displaystyle T\{\cdots \}}$  is the time-ordering operator for which orders the field operators so that earlier time field operators appear to the right of later time field operators. By transforming the fields and states into the interaction picture, this is rewritten as^[1] $${\displaystyle G_{n}(x_{1},\dots ,x_{n})={\frac {\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle }{\langle 0|e^{iS[\phi ]}|0\rangle }},}$$  where ${\displaystyle |0\rangle }$  is the ground state of the free theory and ${\displaystyle S[\phi ]}$  is the action. Expanding ${\displaystyle e^{iS[\phi ]}}$  using its Taylor series, the n-point correlation function becomes a sum of interaction picture correlation functions which can be evaluated using Wick's theorem. A diagrammatic way to represent the resulting sum is via Feynman diagrams, where each term can be evaluated using the position space Feynman rules.

 

A connected Feynman diagram which contributes to the connected six-point correlation function.

 

A disconnected Feynman diagram which does not contribute to the connected six-point correlation function.

The series of diagrams arising from ${\displaystyle \langle 0|e^{iS[\phi ]}|0\rangle }$  is the set of all vacuum bubble diagrams, which are diagrams with no external legs. Meanwhile, ${\displaystyle \langle 0|\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}|0\rangle }$  is given by the set of all possible diagrams with exactly ${\displaystyle n}$  external legs. Since this also includes disconnected diagrams with vacuum bubbles, the sum factorizes into (sum over all bubble diagrams)${\displaystyle \times }$ (sum of all diagrams with no bubbles). The first term then cancels with the normalization factor in the denominator meaning that the n-point correlation function is the sum of all Feynman diagrams excluding vacuum bubbles $${\displaystyle G_{n}(x_{1},\dots ,x_{n})=\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle _{\text{no bubbles}}.}$$ 

While not including any vacuum bubbles, the sum does include disconnected diagrams, which are diagrams where at least one external leg is not connected to all other external legs through some connected path. Excluding these disconnected diagrams instead defines connected n-point correlation functions $${\displaystyle G_{n}^{c}(x_{1},\dots ,x_{n})=\langle 0|T\{\phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}\}|0\rangle _{\text{connected, no bubbles}}}$$ 

It is often preferable to work directly with these as they contain all the information that the full correlation functions contain since any disconnected diagram is merely a product of connected diagrams. By excluding other sets of diagrams one can define other correlation functions such as one-particle irreducible correlation functions.

In the path integral formulation, n-point correlation functions are written as a functional average $${\displaystyle G_{n}(x_{1},\dots ,x_{n})={\frac {\int {\mathcal {D}}\phi \ \phi (x_{1})\dots \phi (x_{n})e^{iS[\phi ]}}{\int {\mathcal {D}}\phi \ e^{iS[\phi ]}}}.}$$ 

They can be evaluated using the partition functional ${\displaystyle Z[J]}$  which acts as a generating functional, with ${\displaystyle J}$  being a source-term, for the correlation functions $${\displaystyle G_{n}(x_{1},\dots ,x_{n})=(-i)^{n}{\frac {1}{Z[J]}}\left.{\frac {\delta ^{n}Z[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}\right|_{J=0}.}$$ 

Similarly, connected correlation functions can be generated using ${\displaystyle W[J]=-i\ln Z[J]}$  as $${\displaystyle G_{n}^{c}(x_{1},\dots ,x_{n})=(-i)^{n-1}\left.{\frac {\delta ^{n}W[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}\right|_{J=0}.}$$ 

Relation to the S-matrix

Scattering amplitudes can be calculated using correlation functions by relating them to the S-matrix through the LSZ reduction formula $${\displaystyle \langle f|S|i\rangle =\left[i\int d^{4}x_{1}e^{-ip_{1}x_{1}}\left(\partial _{x_{1}}^{2}+m^{2}\right)\right]\cdots \left[i\int d^{4}x_{n}e^{ip_{n}x_{n}}\left(\partial _{x_{n}}^{2}+m^{2}\right)\right]\langle \Omega |T\{\phi (x_{1})\dots \phi (x_{n})\}|\Omega \rangle .}$$ 

Here the particles in the initial state ${\displaystyle |i\rangle }$  have a ${\displaystyle -i}$  sign in the exponential, while the particles in the final state ${\displaystyle |f\rangle }$  have a ${\displaystyle +i}$ . All terms in the Feynman diagram expansion of the correlation function will have one propagator for each external leg, that is a propagators with one end at ${\displaystyle x_{i}}$  and the other at some internal vertex ${\displaystyle x}$ . The significance of this formula becomes clear after the application of the Klein–Gordon operators to these external legs using $${\displaystyle \left(\partial _{x_{i}}^{2}+m^{2}\right)\Delta _{F}(x_{i},x)=-i\delta ^{4}(x_{i}-x).}$$ 

This is said to amputate the diagrams by removing the external leg propagators and putting the external states on-shell. All other off-shell contributions from the correlation function vanish. After integrating the resulting delta functions, what will remain of the LSZ reduction formula is merely a Fourier transformation operation where the integration is over the internal point positions ${\displaystyle x}$  that the external leg propagators were attached to. In this form the reduction formula shows that the S-matrix is the Fourier transform of the amputated correlation functions with on-shell external states.

It is common to directly deal with the momentum space correlation function ${\displaystyle {\tilde {G}}(q_{1},\dots ,q_{n})}$ , defined through the Fourier transformation of the correlation function^[2] $${\displaystyle (2\pi )^{4}\delta ^{(4)}(q_{1}+\cdots +q_{n}){\tilde {G}}_{n}(q_{1},\dots ,q_{n})=\int d^{4}x_{1}\dots d^{4}x_{n}\left(\prod _{i=1}^{n}e^{-iq_{i}x_{i}}\right)G_{n}(x_{1},\dots ,x_{n}),}$$  where by convention the momenta are directed inwards into the diagram. A useful quantity to calculate when calculating scattering amplitudes is the matrix element ${\displaystyle {\mathcal {M}}}$  which is defined from the S-matrix via $${\displaystyle \langle f|S-1|i\rangle =i(2\pi )^{4}\delta ^{4}{{\bigg (}\sum _{i}p_{i}{\bigg )}}{\mathcal {M}}}$$  where ${\displaystyle p_{i}}$  are the external momenta. From the LSZ reduction formula it then follows that the matrix element is equivalent to the amputated connected momentum space correlation function with properly orientated external momenta^[3] $${\displaystyle i{\mathcal {M}}={\tilde {G}}_{n}^{c}(p_{1},\dots ,-p_{n})_{\text{amputated}}.}$$ 

For non-scalar theories the reduction formula also introduces external state terms such as polarization vectors for photons or spinor states for fermions. The requirement of using the connected correlation functions arises from the cluster decomposition because scattering processes that occur at large separations do not interfere with each other so can be treated separately.^[4]

Relation to Moments

The two-point function in quantum field theory uses the concept of correlation, which is related to the second moment of the joint distribution of two variables.

In particular, the two-point function describes the correlation between two field operators, which are related to the behavior of particles in the system. It is defined as the vacuum expectation value of the product of two field operators at different points in spacetime:

$${\displaystyle G(x_{1},x_{2})=\langle 0|\phi (x_{1})\phi (x_{2})|0\rangle }$$ 

where ${\displaystyle \phi (x_{1})}$  and ${\displaystyle \phi (x_{2})}$  are the field operators at points ${\displaystyle x_{1}}$  and ${\displaystyle x_{2}}$ , respectively, and ${\displaystyle |0\rangle }$  is the vacuum state.

The two-point function can be written in terms of the covariance between the two field operators, which is related to the second moment of the joint distribution of the two fields. In particular, the covariance between two random variables ${\displaystyle X}$  and ${\displaystyle Y}$  is defined as:

$${\displaystyle {\text{cov}}(X,Y)=\mathbb {E} [(X-\mathbb {E} [X])(Y-\mathbb {E} [Y])]}$$ 

where $\mathbb{E}[X]$ and $\mathbb{E}[Y]$ are the means of $X$ and $Y$, respectively. The covariance measures the extent to which $X$ and $Y$ vary together.

In the case of the two-point function, the field operators ${\displaystyle \phi (x_{1})}$  and ${\displaystyle \phi (x_{2})}$  can be thought of as two random variables that are correlated. The two-point function measures the extent to which the two field operators vary together at different points in spacetime, and it can be written in terms of the covariance between the two operators:

$${\displaystyle G(x_{1},x_{2})={\text{cov}}(\phi (x_{1}),\phi (x_{2}))+{\text{noise}}}$$ 

where the noise term represents the contribution from all other possible correlations between the field operators at different points.

Thus, the two-point function uses the concept of covariance, which is related to the second moment of the joint distribution, to describe the correlation between two field operators and the behavior of particles in the quantum field theory.

Similarly, The three-point function in quantum field theory can be related to statistical moments in a similar way to the two-point function. In particular, the three-point function is related to the third moments of the joint distribution of the field operators.

To see this, consider the expansion of the three-point function in terms of the two-point functions:

$${\displaystyle G(x_{1},x_{2},x_{3})=\langle 0|\phi (x_{1})\phi (x_{2})\phi (x_{3})|0\rangle =G(x_{1},x_{2})G(x_{2},x_{3})G(x_{3},x_{1})+{\text{permutations}}}$$ 

where the permutations refer to all possible ways of permuting the order of the field operators. This formula is known as the Wick's theorem and it provides a way to calculate the three-point function in terms of the two-point functions.

In this formula, the two-point functions ${\displaystyle G(x_{1},x_{2})}$ , ${\displaystyle G(x_{2},x_{3})}$ , and ${\displaystyle G(x_{3},x_{1})}$  can be interpreted as the covariance between the corresponding pairs of field operators. Thus, the three-point function can be written as a product of three covariances, each of which is related to the second moment of the corresponding pair of field operators.

However, the formula also includes additional terms that involve products of three and four two-point functions. These terms involve higher moments of the joint distribution of the field operators, which are related to the third and fourth moments, respectively. In particular, the third moments describe the extent to which the three field operators vary together, while the fourth moments describe the extent to which they vary in pairs.

Thus, the three-point function in quantum field theory is related to statistical moments of the joint distribution of the field operators, in a way that generalizes the relationship between the two-point function and the covariance. This relationship can be used to extract information about the interactions between particles and the underlying structure of the theory from the behavior of the three-point function.

See also

• Effective action
• Green's function (many-body theory)
• Partition function (mathematics)

References

1. Schwartz, M.D. "7". Quantum Field Theory and the Standard Model. Cambridge University Press. ISBN 9781107034730.
2. Năstase, H. (2019). "9". Introduction to Quantum Field Theory. Cambridge University Press. p. 79. ISBN 978-1108493994.
3. Mandl, F.; Shaw, G. (2010). "12". Quantum Field Theory (2 ed.). John Wiley & Sons. p. 254. ISBN 9780471496847.
4. Weinberg, S. (1995). "6". The Quantum Theory of Fields: Foundations. Vol. 1. Cambridge University Press. p. 270. ISBN 9780521670531.

Further reading

• Altland, A.; Simons, B. (2006). Condensed Matter Field Theory. Cambridge University Press.
• Peskin, M.; Schroeder, D.V. (2018) An Introduction to Quantum Field Theory. Addison-Wesley.