Editing Feynman diagram (section)

=== On a lattice ===
On a lattice, (i), the field can be expanded in [[Fourier series|Fourier modes]]:
:<math>\phi(x) = \int \frac{dk}{(2\pi)^d} \phi(k) e^{ik\cdot x} = \int_k \phi(k) e^{ikx}\,.</math>

Here the integration domain is over {{mvar|k}} restricted to a cube of side length {{math|{{sfrac|2π|''a''}}}}, so that large values of {{mvar|k}} are not allowed. It is important to note that the {{mvar|k}}-measure contains the factors of 2{{pi}} from [[Fourier transform]]s, this is the best standard convention for {{mvar|k}}-integrals in QFT. The lattice means that fluctuations at large {{mvar|k}} are not allowed to contribute right away, they only start to contribute in the limit {{math|''a'' → 0}}. Sometimes, instead of a lattice, the field modes are just cut off at high values of {{mvar|k}} instead.

It is also convenient from time to time to consider the space-time volume to be finite, so that the {{mvar|k}} modes are also a lattice. This is not strictly as necessary as the space-lattice limit, because interactions in {{mvar|k}} are not localized, but it is convenient for keeping track of the factors in front of the {{mvar|k}}-integrals and the momentum-conserving delta functions that will arise.

On a lattice, (ii), the action needs to be discretized:
:<math> S= \sum_{\langle x,y\rangle} \tfrac12 \big(\phi(x) - \phi(y) \big)^2\,,</math>

where {{math|{{angbr|''x'',''y''}}}} is a pair of nearest lattice neighbors {{mvar|x}} and {{mvar|y}}. The discretization should be thought of as defining what the derivative {{math|∂<sub>''μ''</sub>''φ''}} means.

In terms of the lattice Fourier modes, the action can be written:
:<math>S= \int_k \Big( \big(1-\cos(k_1)\big) +\big(1-\cos(k_2)\big) + \cdots + \big(1-\cos(k_d)\big) \Big)\phi^*_k \phi^k\,.</math>
For {{mvar|k}} near zero this is:
:<math>S = \int_k \tfrac12 k^2 \left|\phi(k)\right|^2\,.</math>

Now we have the continuum Fourier transform of the original action. In finite volume, the quantity {{mvar|d<sup>d</sup>k}} is not infinitesimal, but becomes the volume of a box made by neighboring Fourier modes, or {{math|<big><big>(</big></big>{{sfrac|2π|''V''}}<big><big>)</big></big>{{su|p=''d''|b=&nbsp;}}}}.

The field {{mvar|φ}} is real-valued, so the Fourier transform obeys:

:<math> \phi(k)^* = \phi(-k)\,.</math>

In terms of real and imaginary parts, the real part of {{math|''φ''(''k'')}} is an [[even function]] of {{mvar|k}}, while the imaginary part is odd. The Fourier transform avoids double-counting, so that it can be written:

:<math> S = \int_k \tfrac12 k^2 \phi(k) \phi(-k)</math>

over an integration domain that integrates over each pair {{math|(''k'',−''k'')}} exactly once.

For a complex scalar field with action

:<math> S = \int \tfrac12 \partial_\mu\phi^* \partial^\mu\phi \,d^dx</math>

the Fourier transform is unconstrained:

:<math> S = \int_k \tfrac12 k^2 \left|\phi(k)\right|^2</math>

and the integral is over all {{mvar|k}}.

Integrating over all different values of {{math|''φ''(''x'')}} is equivalent to integrating over all Fourier modes, because taking a Fourier transform is a unitary linear transformation of field coordinates. When you change coordinates in a multidimensional integral by a linear transformation, the value of the new integral is given by the determinant of the transformation matrix. If
:<math> y_i = A_{ij} x_j\,,</math>

then
:<math>\det(A) \int dx_1\, dx_2 \cdots\, dx_n = \int dy_1\, dy_2 \cdots\, dy_n\,.</math>

If {{mvar|A}} is a rotation, then
:<math>A^\mathrm{T} A = I</math>
so that {{math|det ''A'' {{=}} ±1}}, and the sign depends on whether the rotation includes a reflection or not.

The matrix that changes coordinates from {{math|''φ''(''x'')}} to {{math|''φ''(''k'')}} can be read off from the definition of a Fourier transform.

:<math> A_{kx} = e^{ikx} \,</math>

and the Fourier inversion theorem tells you the inverse:

:<math> A^{-1}_{kx} = e^{-ikx} \,</math>

which is the complex conjugate-transpose, up to factors of 2{{pi}}. On a finite volume lattice, the determinant is nonzero and independent of the field values.
:<math> \det A = 1 \,</math>

and the path integral is a separate factor at each value of {{mvar|k}}.

:<math> \int \exp \left(\frac{i}{2} \sum_k k^2 \phi^*(k) \phi(k) \right)\, D\phi = \prod_k \int_{\phi_k} e^{\frac{i}{2} k^2 \left|\phi_k \right|^2\, d^dk} \,</math>

The factor {{mvar|d<sup>d</sup>k}} is the infinitesimal volume of a discrete cell in {{mvar|k}}-space, in a square lattice box
:<math>d^dk = \left(\frac{1}{L}\right)^d\,,</math>
where {{mvar|L}} is the side-length of the box. Each separate factor is an oscillatory Gaussian, and the width of the Gaussian diverges as the volume goes to infinity.

In imaginary time, the ''Euclidean action'' becomes positive definite, and can be interpreted as a probability distribution. The probability of a field having values {{mvar|φ<sub>k</sub>}} is
:<math> e^{\int_k - \tfrac12 k^2 \phi^*_k \phi_k} = \prod_k e^{- k^2 \left|\phi_k\right|^2\, d^dk}\,. </math>

The expectation value of the field is the statistical expectation value of the field when chosen according to the probability distribution:

:<math>\left\langle \phi(x_1) \cdots \phi(x_n) \right\rangle = \frac{ \displaystyle\int e^{-S} \phi(x_1) \cdots \phi(x_n)\, D\phi} {\displaystyle\int e^{-S}\, D\phi}</math>

Since the probability of {{mvar|φ<sub>k</sub>}} is a product, the value of {{mvar|φ<sub>k</sub>}} at each separate value of {{mvar|k}} is independently Gaussian distributed. The variance of the Gaussian is {{math|{{sfrac|1|''k''<sup>2</sup>''d<sup>d</sup>k''}}}}, which is formally infinite, but that just means that the fluctuations are unbounded in infinite volume. In any finite volume, the integral is replaced by a discrete sum, and the variance of the integral is {{math|{{sfrac|''V''|''k''<sup>2</sup>}}}}.