a � a � where a � and a � are annihilation and creation operators for particles

A ? a ? where a ? and a ? are annihilation and

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a λ - a λ · where a λ and a λ are annihilation and creation operators for particles with wave-fuction e -| ~x | 2 / 2 λ 2 [ a λ , a λ ] = 4 Z e - λ 2 ( | ~ k | 2 + | ~ k 0 | 2 ) / 2 [ a k , a k 0 ] q | ~ k || ~ k 0 | d 3 k d 3 k 0 (2 π ) 3 = Z e - λ 2 | ~ k | 2 | ~ k | d 3 k (2 π ) 3 = 1 4 π 2 λ 4 define A λ = 2 πλ 2 a λ A λ = 2 πλ 2 a λ [ A λ , A λ ] = 1 Q λ (0) = i λ 4 π A λ - A λ · 25
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φ ( x, t ) = Z 1 2 ω k h a k e ikx + a k e - ikx i d 3 k (2 π ) 3 0 φ ( x, t ) = Z 1 2 ω k i k 0 h a k e ikx - a k e - ikx i d 3 k (2 π ) 3 k 0 = ω k = | ~ k | Q λ (0) Z e -| ~ y | 2 / 2 λ 2 0 φ ( ~ y, 0) d 3 y = i λ 3 2 Z e - λ 2 | ~ k | 2 / 2 h a k - a k i q | ~ k | d 3 k (2 π ) 3 / 2 i λ 3 2 a λ - a λ · where a λ and a λ are annihilation and creation operators for particles with wave-fuction e -| ~x | 2 / 2 λ 2 [ a λ , a λ ] = 4 Z e - λ 2 ( | ~ k | 2 + | ~ k 0 | 2 ) / 2 [ a k , a k 0 ] q | ~ k || ~ k 0 | d 3 k d 3 k 0 (2 π ) 3 = Z e - λ 2 | ~ k | 2 | ~ k | d 3 k (2 π ) 3 = 1 4 π 2 λ 4 define A λ = 2 πλ 2 a λ A λ = 2 πλ 2 a λ [ A λ , A λ ] = 1 Q λ (0) = i λ 4 π A λ - A λ · 26
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[ A λ , A λ ] = 1 Q λ (0) = i λ 4 π A λ - A λ · now we want to calculate h 0 | e ic Q λ (0) | 0 i the overlap between the transformed vacuum and the original vacuum d dc h 0 | e ic Q λ (0) | 0 i = i h 0 | Q λ (0) e ic Q λ (0) | 0 i = - h 0 | λ 4 π ( A λ - A λ ) e ic Q λ (0) | 0 i = λ 4 π h 0 | A λ e ic Q λ (0) | 0 i = λ 4 π h 0 | [ A λ , e ic Q λ (0) ] | 0 i now because [ A λ , Q λ (0) ] commutes with Q λ (0) = λ 4 π h 0 | [ A λ , ic Q λ (0) ] e ic Q λ (0) | 0 i = - λ 2 16 π 2 h 0 | [ A λ , c A λ - A λ · ] e ic Q λ (0) | 0 i = - c λ 2 16 π 2 h 0 | e ic Q λ (0) | 0 i ⇒ h 0 | e ic Q λ (0) | 0 i = e - c 2 λ 2 / 32 π 2 the important point is that h 0 | e ic Q λ (0) | 0 i → 0 as λ → ∞ for c fixed 27
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[ A λ , A λ ] = 1 Q λ (0) = i λ 4 π A λ - A λ · now we want to calculate h 0 | e ic Q λ (0) | 0 i the overlap between the transformed vacuum and the original vacuum d dc h 0 | e ic Q λ (0) | 0 i = i h 0 | Q λ (0) e ic Q λ (0) | 0 i = - h 0 | λ 4 π ( A λ - A λ ) e ic Q λ (0) | 0 i = λ 4 π h 0 | A λ e ic Q λ (0) | 0 i = λ 4 π h 0 | [ A λ , e ic Q λ (0) ] | 0 i now because [ A λ , Q λ (0) ] commutes with Q λ (0) = λ 4 π h 0 | [ A λ , ic Q λ (0) ] e ic Q λ (0) | 0 i = - λ 2 16 π 2 h 0 | [ A λ , c A λ - A λ · ] e ic Q λ (0) | 0 i = - c λ 2 16 π 2 h 0 | e ic Q λ (0) | 0 i ⇒ h 0 | e ic Q λ (0) | 0 i = e - c 2 λ 2 / 32 π 2 the important point is that h 0 | e ic Q λ (0) | 0 i → 0 as λ → ∞ for c fixed 28
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[ A λ , A λ ] = 1 Q λ (0) = i λ 4 π A λ - A λ · now we want to calculate h 0 | e ic Q λ (0) | 0 i the overlap between the transformed vacuum and the original vacuum d dc h 0 | e ic Q λ (0) | 0 i = i h 0 | Q λ (0) e ic Q λ (0) | 0 i = - h 0 | λ 4 π ( A λ - A λ ) e ic Q λ (0) | 0 i = λ 4 π h 0 | A λ e ic Q λ (0) | 0 i = λ 4 π h 0 | [ A λ , e ic Q λ (0) ] | 0 i now because [ A λ , Q λ (0) ] commutes with Q λ (0) = λ 4 π h 0 | [ A λ , ic Q λ (0) ] e ic Q λ (0) | 0 i = - λ 2 16 π 2 h 0 | [ A λ , c A λ - A λ · ] e ic Q λ (0) | 0 i = - c λ 2 16 π 2 h 0 | e ic Q λ (0) | 0 i ⇒ h 0 | e ic Q λ (0) | 0 i = e - c 2 λ 2 / 32 π 2 the important point is that h 0 | e ic Q λ (0) | 0 i → 0 as λ → ∞ for c fixed 29
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[ A λ , A λ ] = 1 Q λ (0) = i λ 4 π A λ - A λ · now we want to calculate h 0 | e ic Q λ (0) | 0 i the overlap between the transformed vacuum and the original vacuum d dc h 0 | e ic Q λ (0) | 0 i = i h 0 | Q λ (0) e ic Q λ (0) | 0 i = - h 0 | λ 4 π ( A λ - A λ ) e ic Q λ (0) | 0 i = λ 4 π h 0 | A λ e
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  • Spring '10
  • GEORGI
  • Quantum Field Theory, TA, ........., Quantum chromodynamics, Spontaneous symmetry breaking

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