Prof. Dr. Stefan Floerchinger

Researcher and professor for quantum field theory.

Gaussian field theory

We now consider the \(O(N)\) model in the quadratic regime, i. e. for \(\lambda = 0\). We can write the partition function as \[\begin{split} Z_2[J] = & \int D\phi \, \exp\left( - \frac{1}{2} \int_\mathbf{x} \left\{ \phi_n(\mathbf{x}) (- \partial_j \partial_j + m^2) \phi_n(\mathbf{x}) \right\} + \int_{\mathbf{x}}J_n(\mathbf{x}) \phi_n(\mathbf{x}) \right) \\ = & \int D\phi \, \exp\left( - \frac{1}{2}\int_{\mathbf{x}, \mathbf{y}} \left\{ \phi_n(\mathbf{x}) D_{nm}(\mathbf{x},\mathbf{y}) \phi_m(\mathbf{y}) \right\} + \int_\mathbf{x} J_n(\mathbf{x}) \phi_n(\mathbf{x}) \right) \\ = & \exp\left( \frac{1}{2}\int_{\mathbf{x},\mathbf{y}} J_n(\mathbf{x}) G_{nm}(\mathbf{x},\mathbf{y}) J_m(\mathbf{y}) \right). \end{split}\nonumber\] In the last step we have dropped a constant, i.e. \(J\)-independent, multiplicative factor which is irrelevant for the determination of expectation values and correlation functions. So far this is a formal result, in analogy to Gaussian integration in finite dimensional spaces. We will determine \(G_{nm}(\mathbf{x},\mathbf{y})\) further below.

Field expectation values follow as \[\langle \phi_m(\mathbf{x}) \rangle = \frac{1}{Z_2} \frac{\delta}{\delta J_n(\mathbf{x})} Z_2[J] = \int_\mathbf{y} G_{nm}(\mathbf{x},\mathbf{y}) J_m(\mathbf{y}).\] This is the general result in the presence of a non-vanishing source \(J\). When the latter vanishes this is accordingly also the case for the expectation value.

Two-point functions

Also two-point correlation functions of fields can be calculated within this Gaussian approximation easily, \[\langle \phi_n(\mathbf{x}) \phi_m(\mathbf{y}) \rangle = \frac{1}{Z_2} \frac{\delta^2}{\delta J_n(\mathbf{x}) \delta J_m(\mathbf{y})} Z_2[J] = G_{nm}(\mathbf{x}-\mathbf{y}) + \langle \phi_n(\mathbf{x}) \rangle \langle \phi_m(\mathbf{y}) \rangle.\] The two-point function decomposes into a product of expectation values and a connected correlation function, sometimes also called propagator, \[G_{mn}(\mathbf{x}-\mathbf{y}) = \langle \phi_n(\mathbf{x}) \phi_m(\mathbf{y}) \rangle_c = \langle \phi_n(\mathbf{x}) \phi_m(\mathbf{y}) \rangle - \langle \phi_n(\mathbf{x})\rangle \langle \phi_m(\mathbf{y}) \rangle.\] Usually the connected correlation function goes to zero in the limit of large separation \(|\mathbf{x} - \mathbf{y}| \to 0\).

Greens function

The analog of the matrix \(K_{ij}\) in this field theoretic context is the kernel (read as an infinite-dimensional matrix) \[D_{nm}(\mathbf{x}, \mathbf{y}) = \delta_{nm} \left( - \frac{\partial}{\partial x^j} \frac{\partial}{\partial x^j} + m^2 \right) \delta^{(d)}(\mathbf{x}-\mathbf{y}).\] This is also known as inverse propagator. We need to find its inverse, i. e. another integral operator \(G_{nm}(\mathbf{x},\mathbf{y})\) such that \[\int_\mathbf{y} D_{mn}(\mathbf{x},\mathbf{y}) G_{nk}(\mathbf{y},\mathbf{z}) = \delta_{mk} \delta^{(d)}(\mathbf{x}-\mathbf{z}).\] As a consequence of translational symmetry, \(G_{jk}\) is actually only a function of the difference of coordinates \(\mathbf{x}-\mathbf{y}\). After partial integration we find the relation \[\left( - \frac{\partial}{\partial x^l} \frac{\partial}{\partial x^l} + m^2 \right) G_{jk}(\mathbf{x}-\mathbf{y}) = \delta_{jk} \delta^{(3)}(\mathbf{x}-\mathbf{y}).\] This shows that the so-called propagator \(G_{jk}(\mathbf{x}-\mathbf{y})\) is actually a Greens function to the operator \((-\partial_l^2 + m^2)\). As usual, a Greens function can also depend on the boundary conditions which parametrize here the state of the theory in more detail.

Solution in terms of Fourier transforms

For the ground state one can find the correct Greens function through Fourier transform. We write \[G_{jk}(\mathbf{x}-\mathbf{y}) = \int \frac{d^d p}{(2\pi)^d} e^{i\mathbf{p}(\mathbf{x}-\mathbf{y})} G_{jk}(\mathbf{p}),\] and similarly \[D_{mn}(\mathbf{x}-\mathbf{y}) = \int \frac{d^d p}{(2\pi)^d} e^{i\mathbf{p}(\mathbf{x}-\mathbf{y})} D_{mn}(\mathbf{p}).\] With \(D_{mn}(\mathbf{p}) = \mathbf{p}^2+m^2\) we obtain the simple relation for the Greens function in Fourier space, \[G_{jk}(\mathbf{p}) = \frac{\delta_{jk}}{\mathbf{p}^2+m^2}.\]

Correlation function in position space

For \(d=3\) spatial dimensions, let us calculate the correlation function in position space. The integral can be written as \[\begin{split} G_{jk}(\mathbf{x}-\mathbf{y}) = & \frac{1}{(2\pi)^3}\int d\Omega \int_0^\infty p^2 dp \; e^{ip |\mathbf{x} - \mathbf{y}| \cos(\vartheta)} \frac{\delta_{jk}}{p^2+m^2} \\ = & \frac{4\pi}{2 (2\pi)^3 } \int_{-1}^1 d\cos(\vartheta) \int_0^\infty p^2 dp \; e^{ip |\mathbf{x} - \mathbf{y}| \cos(\vartheta)} \frac{\delta_{jk}}{p^2+m^2} \\ = & \frac{1}{4 i \pi^2 |\mathbf{x}-\mathbf{y}|} \int_0^\infty dp \, p \left(e^{ip |\mathbf{x}-\mathbf{y}| } - e^{-i p|\mathbf{x}-\mathbf{y}|}\right) \frac{\delta_{jk}}{p^2+m^2}. \end{split}\nonumber\] The momentum integral can first be rewritten as an integral along the entire real line and one can then close the integration contour in the upper half of the complex plane, \[\begin{split} G_{jk}(\mathbf{x}-\mathbf{y}) = & \frac{1}{4 i \pi^2 |\mathbf{x}-\mathbf{y}|} \int_{-\infty}^\infty dp \, p \, e^{ip |\mathbf{x}-\mathbf{y}| } \frac{\delta_{jk}}{p^2+m^2} \\ = & \frac{1}{4 i \pi^2 |\mathbf{x}-\mathbf{y}|} \oint dp \, p \, e^{ip |\mathbf{x}-\mathbf{y}| } \frac{\delta_{jk}}{(p+im)(p-im)}. \end{split}\nonumber\] Here one can use the residue theorem which tells that the integral is \(2\pi i\) times the residue at \(p=im\). The final result is \[G_{jk}(\mathbf{x}-\mathbf{y}) = \frac{\delta_{jk}}{4\pi |\mathbf{x}-\mathbf{y}|} e^{-m|\mathbf{x}-\mathbf{y}|}.\] We see that \(m\) has the effect of supressing correlations at large distances exponentially, in addition to an algebraic decay which is also there for \(m=0\). In fact, \[\xi = \frac{1}{m},\] is also known as the correlation length in the context of statistical field theory. We also note that \(G_{jk}(\mathbf{x}-\mathbf{y})\) is divergent in the coincidence limit \(|\mathbf{x}-\mathbf{y}|\to 0\), which corresponds to the region of large wavenumbers in Fourier space. This is known as an ultraviolet divergence. In concrete applications to condensed matter problems there is typically no such divergence but the model theory we have started with looses its physical significance for very high momenta or very short distances.

Exercise

Determine the correlation function \(G_{jk}(x-y)\) in \(d=1\) spatial dimensions. Determine also the four-point correlation function \(\langle \phi(x) \phi(y) \phi(z) \phi(w) \rangle\) for a single real scalar field \(N=1\) in the absense of a source term, i. e. for \(J=0\).

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