From Classical Mechanics to Quantum Field Theory

62 From Classical Mechanics to Quantum Field Theory. A Tutorial

The latter form exhibits explicitly the Moyal product as a series expansion in
powers of. To lowest order:

f∗g=fg+i

2

{f,g}+O

(

^2

)

, (1.328)

where{·,·}is the Poisson bracket. Thus, we see that Planck constantacts
as a “deformation parameter” of the usual associative product structure on the
algebra of functions, making the product non-commutative. Indeed, it can be
seen, e.g., from the expansion of the exponential in Eq.(1.327), that terms pro-
portional to even powers ofare symmetric under the interchangef↔g, but
terms proportional to odd powers areantisymmetric, and this makes the product
non-commutative.

Example 1.3.11.

(1) Iff≡ 1 org≡1, then:

(1∗g)(q,p)=g(q,p),(f∗1) (q,p)=f(q,p). (1.329)

(2) Iff=qand at leastg∈S∞

(

R^2

)

, then:

(q∗g)(q,p)=4

∫

dadbdsdt

(2π)^2 ag(s, t)

×exp

{

2 ı



[(a−q)(t−p)+(s−q)(p−b)]

}

=4

∫

dadbdsdt

(2π)^2

g(s, t)

(

q+i

2

∂

∂t

)

×exp

{

2 ı



[(a−q)(t−p)+(s−q)(p−b)]

}

.

Integrating by parts in the second integral and using the previous result,

one gets:

(q∗g)(q,p)=

(

q+

ı

2

∂

∂p

)

g(q,p). (1.330)

(g∗q)(q,p)=

(

q−

ı

2

∂

∂p

)

g(q,p). (1.331)

(3) In the same way, iff=p,wehave:

(p∗g)(q,p)=

(

p−

ı

2

∂

∂q

)

g(q,p). (1.332)