Editing Feynman diagram (section)

=== Scalar field Lagrangian ===
A simple example is the free relativistic scalar field in {{mvar|d}} dimensions, whose action integral is:
:<math> S = \int \tfrac12 \partial_\mu \phi \partial^\mu \phi\, d^dx \,.</math>

The probability amplitude for a process is:

:<math> \int_A^B e^{iS}\, D\phi\,, </math>

where {{mvar|A}} and {{mvar|B}} are space-like hypersurfaces that define the boundary conditions. The collection of all the {{math|''φ''(''A'')}} on the starting hypersurface give the field's initial value, analogous to the starting position for a point particle, and the field values {{math|''φ''(''B'')}} at each point of the final hypersurface defines the final field value, which is allowed to vary, giving a different amplitude to end up at different values. This is the field-to-field transition amplitude.

The path integral gives the expectation value of operators between the initial and final state:

:<math> \int_A^B e^{iS} \phi(x_1) \cdots \phi(x_n) \,D\phi = \left\langle A\left| \phi(x_1) \cdots \phi(x_n) \right|B \right\rangle\,,</math>

and in the limit that A and B recede to the infinite past and the infinite future, the only contribution that matters is from the ground state (this is only rigorously true if the path-integral is defined slightly rotated into imaginary time). The path integral can be thought of as analogous to a probability distribution, and it is convenient to define it so that multiplying by a constant does not change anything:

:<math> \frac{\displaystyle\int e^{iS} \phi(x_1) \cdots \phi(x_n) \,D\phi }{ \displaystyle\int e^{iS} \,D\phi } = \left\langle 0 \left| \phi(x_1) \cdots \phi(x_n) \right|0\right\rangle \,.</math>

The field's partition function is the normalization factor on the bottom, which coincides with the statistical mechanical partition function at zero temperature when rotated into imaginary time.

The initial-to-final amplitudes are ill-defined if one thinks of the [[continuum limit]] right from the beginning, because the fluctuations in the field can become unbounded. So the path-integral can be thought of as on a discrete square lattice, with lattice spacing {{mvar|a}} and the limit {{math|''a'' → 0}} should be taken carefully{{clarify|date=May 2016}}. If the final results do not depend on the shape of the lattice or the value of {{mvar|a}}, then the continuum limit exists.