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### Transcript of 2 POINTS Homework Assignment #4 16 Homework Assignment #4 Problem 16 Equal time commutation...

• Homework Assignment #4

Problem 16 Equal time commutation relations. For the second quantized Schroedinger equation, we have, in the Schroedinger picture,

[ ψ(x) , ψ☨(x’) ] = δ3(x − x’) , [ ψ(x) , ψ(x’) ] = 0 , etc.

(a ) Show that in the Heisenberg picture, this commutation relation holds at all equal times.

Proof

ψh(x,t) = e i H t ψ(x) e − i H t

ψh✝(x,t) = e i H t ψ✝(x) e − i H t

[ ψh(x,t) , ψh✝(x’,t) ] = [ e i H t ψ(x) e − i H t , e i H t ψ✝(x') e − i H t ]

= e i H t [ ψ(x) , ψ✝(x') ] e − i H t

= δ3(x − x’) which does not depend on t.

(b ) What is the commutation relation for different times?

[ ψh(x,t) , ψh✝(x’,t') ] = [ e i H t ψ(x) e − i H t , e i H t' ψ✝(x') e − i H t' ]

= e i H t ψ(x) e − i H (t − t') ψ✝(x') e − i H t'

− e i H t' ψ✝(x') e + i H (t − t') ψ(x) e − i H t

which is hard to simplify.

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• Problem 17(a) = MANDL and SHAW, problem 2.1

Consider a field theory with a set of fields { φ1(x) , φ2(x) , … , φr(x) , … , φN(x) }.

First consider Lagrangian density ￡ = ￡( φr(x) , ∂αφr(x) ) = ￡( φr , φr,α )

The field equations (i.e., Euler-Lagrangian equations) are

Now modify the Lagrangian density. Let ￡' = ￡ + ∂αΛ

α where Λα (x) is a function of { φ1(x) , φ2(x) , … , φr(x) , … , φN(x) }. Now what are the field equations?

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Note:

17a

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• Problem 17(b) = MANDL and SHAW, problem 2.2 The real scalar field is φ(x) and the Hamiltonian density is

H (x) = ½ { c2 Π2 + (∇φ)2 + μ2 φ2 } The equal-time commutation relations for the quantized fields are

[ φ(x) , Π(x') ] = i ħ δ3( x − x' ) [ φ(x) , φ(x') ] = 0 and [ Π(x) , Π(x') ] = 0

● Calculate [ H , φ(x) ]

● Calculate [ H , Π(x) ]

Next, the Heisenberg equations of motion are

Combining the equations,

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17b

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• Problem 17(c) = MANDL and SHAW, problem 2.3

Consider a real vector field φα(x) with this Lagrangian density ,

￡= − ½ ( ∂α φβ ) ( ∂ α φβ ) + ½ ( ∂α φ

α ) ( ∂β φ β ) + ½ μ2 φα φ

α

or

￡= − ½ ( φβ,α ) ( φ β,α ) + ½ ( φα,α ) ( φ

β ,β ) + ½ μ

2 φα φ α

where φβ,α = ∂α φβ .

● Derive the field equation (Euler-Lagrange equation)

● Also, prove the Lorentz condition: ∂αφ α = 0 .

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17c

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• Problem 18

For the free real scalar field (Section 3.1)

(A ) Write H in terms of Π(x) and φ(x).

(B ) Write H in terms of ak and ak✝ .

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• Problem 19(A) = Mandl and Shaw problem 3.1.

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19A

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• Problem 19(B) = Mandl and Shaw problem 3.2.

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Problem 20 = The Yukawa Theory Problem

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Problem 20 = The Yukawa Theory Problem

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Problem 20 = The Yukawa Theory Problem

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