2.2 From the Source to the Field
2.3.1 Aspects of Functional Calculus
dt Re
Jp(t)qp∗(t)
= 1
2
d4x
J (x)φ†(x) + J†(x)φ(x)
(2.90) So, once again, the physics can be described in terms of a field φ(x) now interacting with an externally specified c-number source.
2.3 Mathematical Supplement
2.3.1 Aspects of Functional Calculus
A functional F (φ) is a mapping from the space of functions to real or com-plex numbers. That is, F (φ) is a rule which allows you to compute a real or complex number for every function φ(x) which itself could be defined, say, in 4-dimensional spacetime. The functional derivative (δF [φ]/δφ(x)) tells you how much the value of the functional (i.e, the value of the computed number) changes if you change φ(x) by a small amount at x. It can be defined through the relation31
δF [φ] =
d4x δF [φ]
δφ(x)δφ(x) (2.91)
Most of the time we will encounter functional derivatives when we vary an integral involving the function φ and the above definition will be the most useful one. One can also define the functional derivatives exactly in analogy with ordinary derivatives. If we change the function by δφ(x) = δ(x− y), then the above equation tells you that
δF [φ] = F [φ + δ(x− y)]−F [φ] =
d4xδF [φ]
δφ(x) δ(x−y) = δF
δφ(y) (2.92) This allows us to define the functional derivative as
δF [φ]
δφ(y) = lim
→0
F [φ + δ(x− y)] − F [φ]
(2.93)
(To avoid ambiguities, we assume that the limit → 0 is taken before any other possible limiting operation.) Note that the right hand side is independent of x in spite of the appearance.
From the above definition one can immediately prove that the product rule of differentiation works for functional derivatives. The chain rule also works, leading to
δ
δφ(y)F [G[φ]] =
d4x δF [G]
δG(x) δG[φ]
δφ(y) (2.94)
From the definition in Eq. (2.93), it is easy to obtain the functional deriva-tives of several simple functionals. For example, we have:
F [φ] =
d4x (φ(x))n; δF [φ]
δφ(y) = n (φ(y))n−1 (2.95) This result can be generalized to any function g[φ(x)] which has a power series expansion, thereby leading to the result
δ δφ(y)
d4x g(φ(x)) = g(φ(y)) (2.96) where prime denotes differentiation with respect to the argument. Next, if x is one dimensional, we can prove:
F [φ] =
dx
dφ(x) dx
n
; δF [φ]
δφ(y) =−n d dx
dφ dx
n−1
y
(2.97)
which again generalizes to any function h[∂aφ] in the form δ
δφ(y)
dx h[∂aφ] =−∂a
∂h
∂ (∂aφ)
y
(2.98)
Very often we will have expressions in which F is a functional of φ as well as an ordinary function of another variable. Here is one example:
F [φ; y] =
d4xK(y, x) φ(x); δF [φ; y]
δφ(x) = K(y, x) (2.99) Finally, note that sometimes F is intrinsically a function of x through its functional dependence on φ like e.g. when F (φ, x) =∇φ(x), or, even more simply, F [φ; x] = φ(x); then the functional derivatives are:
F [φ; x] = φ(x); δF [φ; x]
δφ(y) = δ(x− y) (2.100) and
F (φ, x) =∇φ(x); δFx[φ]
δφ(y) =∇xδ(x− y) (2.101) This is different from the partial derivative ∂F/∂φ which is taken usually at constant∇φ, leading to (∂∇φ/∂φ) = 0. The results in the text can be easily obtained from the above basic results of functional differentiation.
From Fields to Particles
In the previous chapters we obtained the propagation amplitude G(x2; x1) by two different methods. First we computed it by summing over paths in the path integral for a free relativistic particle. Second, we obtained G(x2; x1) by studying how an external c-number source creates and de-stroys particles from the vacuum state. Both approaches, gratifyingly, led to the same expression for G(x2; x1) but it was also clear — in both ap-proaches — that one cannot really interpret this quantity in terms of a single, relativistic particle propagating forward in time. These approaches strongly suggested the interpretation of G(x2; x1) in terms of a system with an infinite number of degrees of freedom, loosely called a field.
In the first approach, we were led (see Eq. (1.131)) to the result:
t,x2
0,x1
exp
−im
t2 t1
dt 1− v2
= G(x2; x1) =0|T [φ(x2)φ(x1)]|0 (3.1)
expressing G(x2; x1) in terms of the vacuum expectation value of the time ordered product of two φs. (For simplicity, we have now assumed that φ is Hermitian.) In the second approach, we were again led (see Eq. (2.54)) to the relation
DJ Z(J) exp
−i
J (x)φ(x)dx
= exp
i
d4x1
2
∂aφ∂aφ− m2φ2 (3.2) which expresses the functional Fourier transform of the vacuum persistence amplitude [Z(J )/Z(0)] = 0+|0−J in the presence of a source J (x), in terms of a functional of a scalar field given by:
A = 1 2
d4x
∂aφ∂aφ− m2φ2
(3.3) We also saw that this functional can be thought of as the action functional for the scalar field (see Eq. (1.146)) and could have been obtained as the sum of the action functionals for an infinite number of harmonic oscillators which were used to arrive at the interpretation of Eq. (3.1). Given the fact that Z(J )∝ 0+|0−J in the left hand side of Eq. (3.2) contains G(x2; x1), this relation again links it to the dynamics of a field φ. More directly, we found that
G(x2; x1) =0|T [φ(x2)φ(x1)]|0 =
Dφ φ(x2)φ(x1) exp (iA[φ]) (3.4)
© Springer International Publishing Switzerland 2016
T. Padmanabhan, Quantum Field Theory, Graduate Texts in Physics, DOI 10.1007/978-3-319-28173-5_3
67
1Some textbooks attempt to “intro-duce” field theory after some mumbo-jumbo about mattresses or springs connected to balls etc. which, if any-thing, confuses the issue. The classi-cal dynamics of a field based on an ac-tion funcac-tional is a trivial extension of classical mechanics based on the action principle, unless you make it a point to complicate it.
thus expressing the propagation amplitude entirely in terms of a path in-tegral average of the fields.
In this chapter we shall reverse the process and obtain the relativistic particle itself as an excitation of a quantum field. In the process, we will re-derive Eq. (3.1) and other relevant results, starting from the dynamics of a field described by the action functionals like the one in Eq. (3.3). Part of this development will be quite straightforward since we have already obtained all these results through one particular route and we only have to reverse the process and make the necessary connections. So we will omit the obvious algebra and instead concentrate on new conceptual issues which crop up.