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

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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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    2 POINTS

  • 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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    18

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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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    19B

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    Problem 20 = The Yukawa Theory Problem

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    2a

    (× A)

    (× A)

    correction

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    2

    Problem 20 = The Yukawa Theory Problem

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    2b

    (× A)

    (× A)

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    868

    868

    Problem 20 = The Yukawa Theory Problem

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    2c

    (× A)

    —— A

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