Pictures Paths Particles Processes

42 November 7, 2013
1.4.3 The classical limit
Since in perturbation theory ¯h is taken to be an infinitesimally small quantity,
the limit ¯h → 0 is of automatic interest. This limit has to be taken with some
care since ¯h = 0 strictly would imply that only Green’s functions with E +L = 1
would survive 24 . Instead, the classical limit ¯h → 0 is meant to be the result of
leaving out diagrams containing closed loops. The diagrammatic SDe will, for
the ϕ 3/4 theory, then take the form
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= +
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+
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. (1.76)
The corresponding solution will be denoted by φ c (J) (with c for ‘classical’), and
the classical SDe is written as
φ c (J) = J µ − λ 3
2µ φ c(J) 2 − λ 4
6µ φ c(J) 3 . (1.77)
The classical field function is exclusively built up from tree diagrams : this is
called the tree approximation. Note that it obeys an algebraic, rather than a
differential, equation, that can be written as
S ′ (φ c (J)) = J . (1.78)
This is called the classical field equation. This is not to be confused with equations
from classical, nonquantum physics. In fact, the classical field equations
will turn out to be the Klein-Gordon, Dirac, Proca and Maxwell equations. Of
these, only the Maxwell equations can be considered classical, since they do not
contain a particle mass.. Note that such equations have, in general, more than
a single solution. Here, however, we are interested in that solution that vanishes
as J → 0, which may be written out using Lagrange expansion :
φ c (J) = J µ + ∑ n≥1
1 ∂ n−1 ( J
[(S
n! µn−1 ′ (∂J) n−1 µ
)) n ]
. (1.79)
Let us now look at the path-integral picture of the classical limit.
becomes small, the fluctuations in the path integrand
(
exp − 1¯h
( ))
S(ϕ) − Jϕ
When ¯h
become extremely exaggerated. The main contribution to 〈ϕ〉 therefore comes
from that value where the probability distribution attains its maximum, that
is,
〈ϕ〉 J
≈ ϕ c , where S ′ (ϕ c ) = J , S ′′ (ϕ c ) > 0 . (1.80)
Also in the classical limit, we therefore have φ c (J) = ϕ c .
24 Later on, the discussion about truncation will clarify how this is not inconsistent.