By Kondratiev A. S., Mazurov V. D.

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49) or, in the index form: G Aaµ = (∂ 2 δ ac + f abc Aµ b ∂µ )ωc . 50) [A] = det[∂ 2 δ ac + f abc Aµ b ∂µ ]−1/2 . 51) So We will discuss strategies to evaluate this determinant shortly. 52) and we can write down Feynman rules. The δ-function remains rather awkward to deal with, though, and this expression can be simpliﬁed through the following trick. Introduce a function, ω, and average over ω with a Gaussian weight factor: Z= [dω]ei d 4 x(ω2 /ξ ) [d Aµ ]δ(G(A) − ω) [A]ei S . 53) We can do the integral over the δ-function.

3) This derivative has the property that it transforms like ψ itself under the symmetry: Dµ ψ → g(x)Dµ ψ. 4) We can also form a gauge-invariant object out of the gauge ﬁelds, Aµ , themselves. A simple way to do this is to construct the commutator of two covariant derivatives: Fµν = i[Dµ , Dν ] = ∂µ Aν − ∂ν Aµ . 5) This form of the gauge transformations may be somewhat unfamiliar. Note, in particular, that the charge of the electron, e (the gauge coupling) does not appear in the transformation laws.

For a continuum ﬁeld theory, states with differing values of the order parameter, v, also have no overlap in the inﬁnite volume limit. This is illustrated by the theory of a scalar ﬁeld with Lagrangian: 1 L = (∂µ φ)2 . 31) 16 2 The Standard Model For this system, there is no potential, so the expectation value, φ = v, is not ﬁxed. 32) is the canonical momentum. So we want to study v|0 = 0|ei Q |0 . 33) We must be careful how we take the inﬁnite-volume limit. We will insist that this be done in a smooth fashion, so we will deﬁne: Q= = −i d 3 x∂0 φe−x d 3k (2π )3 2 /V 2/3 ωk 2 V 1/3 √ π 3 e−k 2 V 2/3 /4 [a(k) − a † (k)].

### 2-Signalizers of Finite Simple Groups by Kondratiev A. S., Mazurov V. D.

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