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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.<ref>{{cite book|first=S.|last=Weinberg| author1-link=Steven Weinberg|title=The Quantum Theory of Fields: Foundations| publisher=Cambridge University Press| date=1995| chapter=6|volume=1| page=270| isbn=9780521670531}}</ref>
== Relation to [[Moment_(mathematics)|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:
<math display="block"> G(x_1,x_2) = \langle 0| \phi(x_1) \phi(x_2) |0 \rangle </math>
where <math>\phi(x_1)</math> and <math>\phi(x_2)</math> are the field operators at points <math>x_1</math> and <math>x_2</math>, respectively, and <math>|0\rangle</math> 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 <math>X</math> and <math>Y</math> is defined as:
<math display="block">\text{cov}(X,Y) = \mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]</math>
where $\mathbb{E}[X]$ and $\mathbb{E}[Y]$ are the means of <math>X</math> and <math>Y</math>, respectively. The covariance measures the extent to which $X$ and $Y$ vary together.
In the case of the two-point function, the field operators <math>\phi(x_1)</math> and <math>\phi(x_2)</math> 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:
<math display="block"> G(x_1,x_2) = \text{cov}(\phi(x_1),\phi(x_2)) + \text{noise} </math>
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:
<math display="block">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}</math>
where the permutations refer to all possible ways of permuting the order of the field operators. This formula is known as the [[Wick%27s_theorem|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 <math>G(x_1,x_2)</math>, <math>G(x_2,x_3)</math>, and <math>G(x_3,x_1)</math> 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==
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