From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 197

mi;i=1, 2 ,···,n. The position and momentum operators are given in the
Schr ̈odinger representation by

xˆiψ(x)=xiψ(x); pˆiψ(x)=−i∂iψ(x),

wherex=(x 1 ,x 2 ,...,xn)∈Rn. They satisfy the canonical commutation rela-
tions

[ˆxi,pˆj]=iδij.

The Hamiltonian

Hˆ=−

∑n

i=1

1

2 mi

(

∂^2 i+m^2 iωi^2 x^2 i

)

,

in terms of the creation and destruction operators

ai=

1

√

2 miωi

(∂i+miωixi); a†=

1

√

2 miωi

(−∂i+miωixi),

reads

Hˆ=

∑n

i=1

ωi

(

a†iai+^1

2

)

The ground state

〈x| 0 〉=

(n

∏

i=1

√

ωi

π

)

e−

12 ∑ni=1ωix (^2) i
,
has an energy given by the sum of the ground state energies of the different har-
monic modes
E 0 =

1

2

∑n

i=1

ωi.

In a similar way, it is easy to check that the excited states Eq. (3.8) are given
by

〈x|n 1 ,n 2 ,...,nn〉=

(n

∏

i=1

Hni(xi)

)

e−

12 ∑ni=1ωix (^2) i
,
in terms of the Hermite polynomialsHni,i=1, 2 ,...,n.

3.2.2 Relativity and the Poincar ́egroup.............

The Einstein theory of Relativity is based on the unification of space and time
into a four-dimensional space-timeR^4 equipped with the Minkowski metric

dx^2 =dx^20 −dx^21 −dx^22 −dx^23 =

∑^3

μ=0

ημνdxμdxν,