Editing Feynman diagram (section)

=== Combining denominators ===
The Schwinger representation has an immediate practical application to loop diagrams. For example, for the diagram in the {{math|''φ''<sup>4</sup>}} theory formed by joining two {{mvar|x}}s together in two half-lines, and making the remaining lines external, the integral over the internal propagators in the loop is:

: <math> \int_k \frac{1}{k^2 + m^2} \frac{1}{ (k+p)^2 + m^2} \,.</math>

Here one line carries momentum {{mvar|k}} and the other {{math|''k'' + ''p''}}. The asymmetry can be fixed by putting everything in the Schwinger representation.

:<math> \int_{t,t'} e^{-t(k^2+m^2) - t'\left((k+p)^2 +m^2\right) }\, dt\, dt'\,. </math>

Now the exponent mostly depends on {{math|''t'' + ''t''′}},

:<math> \int_{t,t'} e^{-(t+t')(k^2+m^2) - t' 2p\cdot k -t' p^2}\,, </math>

except for the asymmetrical little bit. Defining the variable {{math|''u'' {{=}} ''t'' + ''t''′}} and {{math|''v'' {{=}} {{sfrac|''t''′|''u''}}}}, the variable {{mvar|u}} goes from 0 to {{math|∞}}, while {{mvar|v}} goes from 0 to 1. The variable {{mvar|u}} is the total proper time for the loop, while {{mvar|v}} parametrizes the fraction of the proper time on the top of the loop versus the bottom.

The [[Jacobian matrix and determinant#Jacobian determinant|Jacobian]] for this transformation of variables is easy to work out from the identities:

:<math> d(uv)= dt'\quad du = dt+dt'\,,</math>

and "[[exterior product|wedging]]" gives

:<math> u\, du \wedge dv = dt \wedge dt'\,</math>.

This allows the {{mvar|u}} integral to be evaluated explicitly:

:<math> \int_{u,v} u e^{-u \left( k^2+m^2 + v 2p\cdot k + v p^2\right)} = \int \frac{1}{\left(k^2 + m^2 + v 2p\cdot k - v p^2\right)^2}\, dv </math>

leaving only the {{mvar|v}}-integral. This method, invented by Schwinger but usually attributed to Feynman, is called ''combining denominator''. Abstractly, it is the elementary identity:

:<math> \frac{1}{AB}= \int_0^1 \frac{1}{\big( vA+ (1-v)B\big)^2}\, dv </math>

But this form does not provide the physical motivation for introducing {{mvar|v}}; {{mvar|v}} is the proportion of proper time on one of the legs of the loop.

Once the denominators are combined, a shift in {{mvar|k}} to {{math|''k''′ {{=}} ''k'' + ''vp''}} symmetrizes everything:

:<math> \int_0^1 \int\frac{1}{\left(k^2 + m^2 + 2vp \cdot k + v p^2\right)^2}\, dk\, dv = \int_0^1 \int \frac{1}{\left(k'^2 + m^2 + v(1-v)p^2\right)^2}\, dk'\, dv</math>

This form shows that the moment that {{math|''p''<sup>2</sup>}} is more negative than four times the mass of the particle in the loop, which happens in a physical region of [[Lorentz space]], the integral has a cut. This is exactly when the external momentum can create physical particles.

When the loop has more vertices, there are more denominators to combine:

:<math> \int dk\, \frac{1}{k^2 + m^2} \frac{1}{(k+p_1)^2 + m^2} \cdots \frac{1}{(k+p_n)^2 + m^2}</math>

The general rule follows from the Schwinger prescription for {{mvar|''n'' + 1}} denominators:

:<math> \frac{1}{D_0 D_1 \cdots D_n} = \int_0^\infty \cdots\int_0^\infty e^{-u_0 D_0 \cdots -u_n D_n}\, du_0 \cdots du_n \,.</math>

The integral over the Schwinger parameters {{mvar|u<sub>i</sub>}} can be split up as before into an integral over the total proper time {{math|''u'' {{=}} ''u''<sub>0</sub> + ''u''<sub>1</sub> ... + ''u<sub>n</sub>''}} and an integral over the fraction of the proper time in all but the first segment of the loop {{math|''v<sub>i</sub>'' {{=}} {{sfrac|''u<sub>i</sub>''|''u''}}}} for {{math|''i'' ∈ {{mset|1,2,...,''n''}}}}. The {{mvar|v<sub>i</sub>}} are positive and add up to less than 1, so that the {{mvar|v}} integral is over an {{mvar|n}}-dimensional simplex.

The Jacobian for the coordinate transformation can be worked out as before:

:<math> du = du_0 + du_1 \cdots + du_n \,</math>
:<math> d(uv_i) = d u_i \,.</math>

Wedging all these equations together, one obtains

:<math> u^n\, du \wedge dv_1 \wedge dv_2 \cdots \wedge dv_n = du_0 \wedge du_1 \cdots \wedge du_n \,.</math>

This gives the integral:

:<math> \int_0^\infty \int_{\mathrm{simplex}} u^n e^{-u\left(v_0 D_0 + v_1 D_1 + v_2 D_2 \cdots + v_n D_n\right)}\, dv_1\cdots dv_n\, du\,, </math>

where the simplex is the region defined by the conditions
:<math>v_i>0 \quad \mbox{and} \quad \sum_{i=1}^n v_i < 1 </math>
as well as
:<math>v_0 = 1-\sum_{i=1}^n v_i\,.</math>
Performing the {{mvar|u}} integral gives the general prescription for combining denominators:

:<math> \frac{1}{ D_0 \cdots D_n } = n! \int_{\mathrm{simplex}} \frac{1}{ \left(v_0 D_0 +v_1 D_1 \cdots + v_n D_n\right)^{n+1}}\, dv_1\, dv_2 \cdots dv_n </math>

Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs. The interpretation of the parameters {{mvar|v<sub>i</sub>}} is that they are the fraction of the total proper time spent on each leg.