e ik(x y) m 2 where the last equality follows from eq. (3.10a) for the sum over polarizations. = i
|
|
- Norah Chase
- 5 years ago
- Views:
Transcription
1 PHY 396 K. Solutions for homework set #4. Problem 1a): Each quantum field is a linear combination 3.11) of creation and annihilation operators. The product of two fields µ x)âν y) therefore decomposes into products of two creation or annihilation operators, ââ, â â, â â, or ââ. The first three types of products have zero matrix elements between two vacuum states, 0 â k,λ â k,λ 0 = 0 â k,λâ k,λ 0 = 0 â k,λâk,λ 0 = 0 k, k, λ, λ, S.1) while the fourth type has non-zero matrix element when k = k and λ = λ, 0 â k,λ â k,λ 0 = 2ω k 2π) 3 δ 3) k k) δ λλ. S.2) Consequently, 0 µ x)âν y) 0 = = = d 3 k 1 d 3 k 1 2π) 3 2ω k 2π) 3 2ω k e ikx e +ik y E µ k,λ Eν k,λ 0 ] â k,λâk,λ 0 λ,λ d 3 k 1 ] k 2π) 3 e ikx e +iky E µ 0 =+ω k 2ω k,λ Eν k,λ k d 3 k 2π) 3 1 2ω k λ e ikx y) g µν + kµ k ν m 2 + three terms that vanish )] k 0 =+ω k where the last equality follows from eq. 3.10a) for the sum over polarizations. Moreover, S.3) d 3 k 2π) 3 1 2ω k e ikx y) k µ] k 0 =+ω k and likewise d 3 k 1 = i x µ 2π) 3 e ikx y)] k 0 =+ω k 2ω k S.4) d 3 k 2π) 3 1 2ω k e ikx y) k µ k ν] k 0 =+ω k = x µ x ν d 3 k 2π) 3 1 2ω k e ikx y)] k 0 =+ω k, S.5) 1
2 which allows us to rewrite the last line of eq. S.3) as 0 µ x)âν y) 0 = = g µν 1m 2 2 x µ x ν g µν 1m 2 2 x µ x ν ) d 3 k 2π) 3 1 2ω k e ikx y)] k 0 =+ω k ) Dx y). S.6) Q.E.D. Problem 1b): In light of eq. 3.18) for the vacuum sandwich, we immediately see that for the un-modified time-ordering, 0 Tµ x)âν y) 0 = Θx 0 y 0 ) 0 µ x)âν y) 0 + Θy 0 x 0 ) 0 Âν y)âµ x) 0 On the other hand, we saw in class that = Θx 0 y 0 ) g µν 1 ) m 2 Dx y) x µ x ν + Θy 0 x 0 ) g µν 1 ) m 2 Dy x). x µ x ν S.7) G scalar F x y) = Θx 0 y 0 )Dx y) + Θy 0 x 0 )Dy x) S.8) and also G scalar F x y) = Θx 0 y 0 Dx y) ) + Θy 0 x 0 Dy x) ) x 0 x 0 x 0 S.9) but 2 x 2 0 G scalar F x y) = Θx 0 y 0 ) 2 Dx y) x Θy 0 x 0 ) 2 Dy x) x 2 0 iδ 4) x y). S.10) Since the space derivatives commute with the Θ functions of times, eqs. S.8) through S.10) 2
3 imply x µ xg ν scalar F x y) = Θx 0 y 0 ) x µ xdx y) ν + Θy 0 x 0 ) x µ xdy x) ν iδ µ0 δ ν0 δ 4) x y) and therefore S.11) g µν µ x x ν ) m 2 G scalar F x y) = Θx 0 y 0 ) g µν µ x x ν m 2 in light of eq. S.7) + Θy 0 x 0 ) + iδµ0 δ ν0 m 2 δ 4) x y) g µν µ x ν x m 2 = 0 Tµ x)âν y) 0 + iδµ0 δ ν0 0 T  µ x)âν y) 0 ) Dx y) ) Dy x) m 2 δ 4) x y) S.12) where the last equality is the definition 3.20) of the modified time-ordered product of vector fields. The above proves the first line of eq. 3.19); the second line follows from the momentumspace form of the scalar propagator: d G scalar 4 k F x y) = 2π) 4 ie ikx y) k 2 m 2 + i0, S.13) hence G µν F x y) = g µν µ x x ν ) d m 2 G scalar 4 k F x y) = 2π) 4 Q.E.D. g µν + kµ k ν m 2 ) ie ikx y) k 2 m 2 + i0. S.14) 3
4 Problem 1c): The classical Lagrangian of the free massive vector field is L = 1 4 F νµf νµ + 2 1m2 A µ A µ = 1 2 νa µ ) ν A µ µ A ν ) m2 A µ A µ S.15) = 1 2 A µ 2 A µ + m 2 A µ µ ν ) A ν + a total derivative. Integrating this Lagrangian gives us the classical action as S = d 4 x L = 1 2 d 4 x A µ x) D µν A ν x) S.16) where D µν is the second-order differential operator This verifies eq. 3.21). D µν = 2 + m 2 ) g µν µ ν. S.17) Now let s apply the operator S.17) to the Feynman propagator 3.19) of the vector fields. Using the top line of eq. 3.19) relating G F νλ to scalar propagator, we obtain D µν G F νλ x y) = 2 + m 2 ) g µν µ ν) g νλ 1 ) m 2 ν λ G scalar F x y) = δ µ λ 2 + m 2 ) + µ λ 2 + m 2 m 2 µ λ + µ 2 m 2 λ Gscalar F x y) = δ µ λ 2 + m 2 ) + 0 µ λ S.18) = δ µ λ 2 + m 2 )G scalar F x y). But the scalar propagator is the Green s function of the Klein Gordon operator 2 + m 2 ), which immediately gives us 2 + m 2 )G scalar F x y) = iδ 4) x y), S.19) D µν G F νλ x y) = +iδµ λ δ4) x y). S.20) In other words, the vector propagator is the Green s function of the differential operator D µν, Q.E.D. 4
5 Problem 3a): Ĵi, Ĵj] 1 4 ɛikl ɛ jmn Ĵ kl, Ĵmn] = by eq. 1) = 1 4 ɛikl ɛ jmn ig km Ĵ ln + ig kn Ĵ lm + ig lm Ĵ kn g ln Ĵ km) by antisymmetry of ɛ s = ɛ ikl ɛ jmn ig km Ĵ ln = iĵln g km ɛ ikl ɛ jmn = +δ km ɛ ikl ɛ jmn = δ ij δ ln δ in δ lj) = 0 iĵji = +iĵij +iɛ ijk Ĵ k. S.21) Ĵi, ˆK j] 1 2 ɛikl Ĵ kl, Ĵ0j] = by eq. 1) = 1 2 ɛikl ig k0 Ĵ lj + ig kj Ĵ l0 + ig l0 Ĵ kj ig lj Ĵ k0) = 1 2 ɛikl 0 iδ kj Ĵ l iδ lj Ĵ k0) 1 2 ɛikl +iδ kj ˆKl iδ lj ˆKk ) = 1 2 ɛijl ˆKl 1 2 ɛikj ˆKj ˆKi, ˆK j] Ĵ0i, Ĵ0j] = by eq. 1) = ɛ ijk ˆKk, S.22) = ig 00 Ĵ ij + ig 0j Ĵ i0 + ig i0 Ĵ 0j ig ij Ĵ 00 = iĵij , iɛ ijk Ĵ k. S.23) Problem 3b): ˆV i, Ĵj] 1 2 ɛjkl ˆV i, Ĵkl] = 1 2 ɛjkl ig ik ˆV l ig il ˆV k ) = 1 2 ɛjkl iδ ik ˆV l + δ il ˆV k ) = i 2 ɛjil ˆV l + i 2 ɛjki ˆV k = iɛ ijk ˆV k, S.24) 5
6 ˆV 0, Ĵj] = 1 2 ɛjkl ˆV 0, Ĵkl] = 1 2 ɛjkl i\g 0k ˆV l i\g 0l ˆV k ) = 0, S.25) ˆV i, ˆK j] = ˆV i, Ĵ0j] = i\g i0 ˆV j ig ij ˆV 0 = +iδ ij ˆV 0, S.26) ˆV 0, ˆK j] = ˆV 0, Ĵ0j] = ig 00 ˆV j i\g 0j ˆV 0 = +i ˆV j. S.27) Note that the Hamiltonian of a relativistic theory is a member of a 4-vector multiplet ˆP µ = Ĥ, ˆP) where ˆP is the net momentum operator. Applying the above equations to the ˆP µ vectors, we obtain ˆP i, Ĵj] = iɛ ijk ˆP k, Ĥ, Ĵ j] = 0, ˆP i, ˆK j] = +iδ ij Ĥ, S.28) Ĥ, ˆKj ] = +i ˆP j. In particular, the Hamiltonian Ĥ commutes with the three angular momenta Ĵj but it does not commute with the three generators ˆK k of the Lorentz boosts. Comment: In the ordinary quantum mechanics, it is often said that generators of continuous symmetries must commute with the Hamiltonian operator. However, this is true only for the symmetries that act in a time independent manner for example, rotating the 3D space by the same angle at all times t. But when the transformation rules of a symmetry depend on time, the Hamiltonian must change to account for this time dependence. In a Lorentz boost, the transform x x obviously depends on time, which changes the way the transformed quantum fields such as ˆΦ x, t) depend on t. Consequently, the Hamiltonian Ĥ of the theory must change so that the new Heisenberg equations would match the new time dependence. In terms of the generators, this means that the boost generators ˆK i should not commute with the Hamiltonian. 6
7 Note that this non-commutativity is not caused by Lorentz boosts affecting the time itself, t = L 0 µx µ t. Even in non-relativistic theories where the time is absolute generators of symmetries that affect the other variables in a time-dependent matter do not commute with Ĥ. For example, consider a Galilean transform from one non-relativistic moving frame into another, x = x + vt but t = t. This is a good symmetry of non-relativistic particles interacting with each other but not subject to any external potential, Ĥ = a 1 2M ˆp2 a V ˆx a ˆx b ). a b S.29) A unitary operator operators as Ĝ realizing a Galilean symmetry acts on coordinate and momentum Ĝˆx a Ĝ = ˆx a + vt, Ĝˆp a Ĝ = ˆp a + Mv, S.30) and it also transforms the Hamiltonian into ĜĤĜ = Ĥ + v ˆP tot M totv 2. S.31) In terms of the Galilean boost generators ˆK G, Ĝ = exp iv ˆK G ), ˆx i a, ˆK j G ] = iδ ij t, ˆp i a, ˆK ] j G In particular, the ˆK G i do not commute with the Hamiltonian. = imδ ij, ] Ĥ, ˆKj G = i ˆP i tot. S.32) Problem 3c): Consider a linear combination 1 2 N µνĵµν of Lorentz generators with some generic coefficients N µν = N νµ. The Lorentz symmetries LN, φ) = exp iφ 2 N µνĵµν ) generated by this combination preserve the momentum p µ of the particle state p if and only if exp iφ 2 N µνĵµν) ˆP α exp iφ 2 N µνĵµν) p = ˆP µ p. S.33) 7
8 For an infinitesimal φ, this condition becomes iφ 2 N µνĵµν, ˆP α] p = 0. S.34) Applying eqs. 3) to the left hand side of this formula gives us iφ 2 N µνĵµν, ˆP α] p = iφ 2 N µν ig αµ ˆP ν + g αν ˆP µ ) p = φn αν ˆPν p S.35) = φn αν p ν p, so the condition for the generator 1 2 N µνĵµν to preserve particle s momentum p µ is simply N αν p ν = 0. S.36) In 3D terms, N ij = ɛ ijk a k and N 0k = N k0 = b k for some 3-vectors a and b, the generator in question is 1 2 N µν Ĵ µν = a Ĵ + b ˆK, S.37) and the condition S.36) becomes a p be = 0 and b p = 0. S.38) Actually, the second condition here is redundant, so the general solution is any a, b = a p E. S.39) In terms of eq. S.37), these solutions mean 1 2 N µν Ĵ µν = a Ĵ + a p) ˆK E = a p ) Ĵ + E ˆK for any a. S.40) In other words, the Lorentz symmetries preserving the momentum p µ have 3 generators, 8
9 namely the components of the 3-vector Ĵ + p E ˆK S.41) For a particle moving in z direction at speed β = p z /E, these components are Ĵ x β ˆK y, Ĵ y + β ˆK x, and Ĵ z. S.42) Naturally, the generators of any symmetry are defined up to linear combinations. Thus, any 3 linearly-independent combination of the operators S.42) will generate the same little group Gp) of Lorentz symmetries preserving the momentum p µ. In particular, in eq. 4) I have multiplied 2 of the generators S.42) by the Lorentz slowdown factor γ while leaving Ĵ z as it is. The purpose of this rescaling is to make all 3 generators normalized as angular momenta, see part d) below. Problem 3d): Ĵz, J x] Ĵz Ĵz = γ, Ĵx] βγ, ˆK y] = γ iĵy βγ i ˆK x ) = i J y, Ĵz, J y] Ĵz Ĵz = γ, Ĵy] + βγ, ˆK x] = γ iĵx ) βγ +i ˆK y ) S.43) = i J x, Jx, J y] = γ 2 Ĵ x, Ĵy] βγ 2 ˆKy, Ĵy] + βγ 2 Ĵ x, ˆK x, ] = γ 2 iĵz β 2 γ 2 iĵz = iĵz γ 2 1 β 2 ) = 1 ) = iĵz. β 2 γ 2 ˆKy, ˆK x] 9
10 Problem 3, comment to eq. 5): For a massless particle, the generators of the little group of its momentum are exactly as in eq. S.42) for β = 1 a massless particle moves at the speed of light). Indeed, the analysis of part c) above goes through without any changes, except we cannot rescale two of the generators by a factor of γ because a massless particle has γ =. Without the γ factors, the commutation relations Ĵz, Îx ] = iîy and Ĵz, Îy ] = iîx work exactly as in the first two eqs. S.43), but in the third equation we get Îx, Îy] = iĵz 1 β 2 ) without the γ 2 factor = 0. S.44) Problem 3e): The quantum state p, λ has definite momentum p µ, thus ˆP α p, λ = p α p, λ and likewise ɛ αβγδ Ĵ βγ ˆP δ p, λ = ɛ αβγδ p δ Ĵ βγ p, λ. S.45) For simplicity, let s assume the particle moves in the z direction and spell out the operator ˆQ α = ɛ αβγδ p δ Ĵ βγ in components: ˆQ 0 = ɛ 0ij3 p 3 Ĵij = 2p 3 Ĵ12 = 2p z J z, ˆQ 1 = ɛ 1023 p 3 Ĵ02 + ɛ 1203 p 3 Ĵ20 + ɛ 1jk0 p 0 Ĵjk = 2p 3 Ĵ02 2p 0 Ĵ23 = 2p z ˆK y 2p 0 Ĵx, ˆQ 2 = ɛ 2013 p 3 Ĵ01 + ɛ 2103 p 3 Ĵ10 + ɛ 2jk0 p 0 Ĵjk S.46) = +2p 3 Ĵ01 2p 0 Ĵ31 = +2p z ˆK x 2p 0 Ĵy, ˆQ 3 = ɛ 3ij0 p 0 Ĵij = 2p 0 Ĵ12 = 2p 0 Ĵ z. For a massless particle with p z = p 0 = E, these components simplify to ˆQ 0 = +2E Ĵz, ˆQx = 2E Îx, ˆQy = 2E Îy, ˆQz = 2E Ĵz. S.47) Besides definite momentum, the state p, λ has definite helicity λ, Ĵ z p, λ = λ p, λ. Moreover, it is annihilated by other two generators of the little group of its momentum, 10
11 Î x p, λ = Îy p, λ = 0. Therefore, when we apply the operators S.47) to this state, we obtain ˆQ 0 p, λ = +2Eλ p, λ, ˆQ x p, λ = 0, ˆQ y p, λ = 0, S.48) ˆQ z p, λ = 2Eλ p, λ, Comparing the right hand sides here to the particle momentum with a lower index) p α = +E, 0, 0, E) we see that ˆQ α p, λ = 2λp α p, λ. S.49) In other words, ɛ αβγδ Ĵ βγ ˆP δ p, λ = 2λ ˆP α p, λ. 6) become What if a massless particle moves in another direction? In 3-vector notations, eqs. S.46) ˆQ 0 = 2E β Ĵ, ˆQ = 2E Ĵ + β ˆK ). S.50) Up the overall factor 2E, the components of ˆQ are the generators S.42) of the little group of the momentum p µ. A singlet multiplets of this little group is annihilated by the two generators p and is an eigenstate of the third generator p. Indeed, for a massless particle β Ĵ is the helicity operator, hence ˆQ 0 p, λ = 2Eλ p, λ, β ˆQ p, λ = 2E β Ĵ p, λ = 2Eλ p, λ, S.51) while β ˆQ p, λ = 0. S.52) Consequently, ˆQ p, λ = β ) β ˆQ p, λ β β ˆQ p, λ = β 2Eλ p, λ ) = 2pλ p, λ, S.53) 11
12 and combining this formula with ˆQ 0 p, λ = 2Eλ p, λ we obtain ˆQ µ p, λ = 2λp µ p, λ. S.54) This proves eq. 6) for a massless particle moving in any direction. Now consider a continuous Lorentz transform x µ x µ = L µ νx ν. The operators on both sides of both sides of eq. 6) transform as Lorentz vectors, ˆDL) ˆP α ˆD L) = L µ α ˆP µ, ˆDL) ɛ αβγδ Ĵ βγ ˆP δ ) ˆD L) = L µ α ɛ µβγδ Ĵ βγ ˆP δ ). S.55) Consequently, the transformed state ˆD p, λ = Lp,?? S.56) satisfies the same eq. 6) as the original state p, λ. Indeed, L α µ ɛ µβγδ Ĵ βγ δ ˆP ) Lp,?? = ˆDL) ɛ αβγδ Ĵ βγ δ ˆP ) ˆD L) ˆD p, λ ɛ αβγδ Ĵ βγ δ ˆP ) p, λ = ˆDL) by eq. 6) = ˆDL) 2λ ˆP α p, λ S.57) and hence = 2λ ˆDL) ˆP α ˆD L) ˆD p, λ = 2λ L µ α ˆP µ Lp,??, ɛ µβγδ Ĵ βγ ˆP δ ) Lp,?? = 2λ ˆP µ Lp,??. S.58) Note that this equation for the transformed state Lp,?? has exactly the same helicity eigenvalue λ as the original eq. 6). Physically, this means that the transformed state has the same helicity as the original state for massless particles, continuous Lorentz transforms preserve helicity! 12
13 Problem 4a): In light of eqs. 2), Ĵi ±, Ĵj ±] = 1 4 Ĵi, Ĵj] ± 4 i ˆKi, Ĵj] ± 4 i Ĵi, ˆK j] 4 1 ˆKi, ˆK j] = 1 4 iɛijk Ĵ k ± i 4 iɛijk ˆKk ± i 4 iɛijk ˆKk 1 4 i)ɛijk Ĵ k = ɛ ijk i 2 Ĵ k 1 2 ˆK k) = iɛ ijk Ĵ k ±, Ĵi ±, Ĵj ] = 1 4 Ĵi, Ĵj] ± 4 i ˆKi, Ĵj] 4 i Ĵi, ˆK j] ˆKi, ˆK j] S.59) = 1 4 iɛijk Ĵ k ± i 4 iɛijk ˆKk i 4 iɛijk ˆKk i)ɛijk Ĵ k = 0. Problem 4b): First of all, for any complex matrix A, A n) = A ) n and hence for any function f that expands into a power series with real coefficients, fa) ) = f A ). In particular, expa) ) = exp A ) = expa) ) ) 1 = exp A ). S.60) Now, in eq. 8) for M, the 3-vectors a and b are real and the Pauli matrices σ x,y,z are hermitian. So if we let A = 1 2 ia + b) σ, S.61) then A = 1 2 ia b) σ. S.62) Plugging these A and A into eq. S.60) and comparing to eqs. 8) and 9) we immediately obtain M = expa), M = exp A ) = M ) 1 = M. 10a) To prove the second equality in eq. 10) we need the Pauli-matrix relation σ 2 σ σ 2 = σ, S.63) which follows from σ 2 being imaginary antisymmetric matrix while the σ 1 and σ 3 matrices 13
14 are real and symmetric but anticommute with the σ 2. Indeed, matrix by matrix, σ 2 σ 1σ 2 = +σ 2 σ 1 σ 2 = σ 2 σ 2 σ 1 = σ 1, σ 2 σ 2σ 2 = σ 2 σ 2 σ 2 = σ 2 σ 2 σ 2 = σ 2, S.64) σ 2 σ 3σ 2 = +σ 2 σ 3 σ 2 = σ 2 σ 2 σ 3 = σ 3. Thanks to the relations S.63) and to σ 2 σ 2 = 1 for any complex vector c, σ 2 c σ) n) σ2 = σ 2 c σ ) σ 2 ) n = c σ) n and hence for any any function f that expands into a power series with real coefficients, σ 2 fc σ) ) σ2 = f c σ). In particular, σ 2 exp 12 ia + b) σ )) σ2 = exp 1 2 ia + b) σ ) = exp 1 2 ia b) σ ), S.65) i.e., σ 2 M σ 2 = M. Q.E.D. Problem 4c): Let s start with reality. The matrix V = V µ σ µ is hermitian if and only if the 4-vector V µ is real. For any matrix M SL2, C), the transform V V = MV M S.66) preserves hermiticity: if V is hermitian, then so is V ; indeed V ) = MV M ) = M ) V M = MV M = V. S.67) In terms of the 4-vectors, this means that if V µ is real than V µ = L µ νv ν is also real. In other words, the 4 4 matrix L µ νm) is real. 14
15 Next, let s prove that L µ νm) O3, 1) it preserves the Lorentz metric g αβ, or equivalently, for any V µ, g αβ V αv β = gαβ V α V β. In 2 2 matrix terms, the Lorenz square of a 4-vector becomes the determinant: g αβ V α V β = det V = V µ σ µ). S.68) Indeed, from the explicit form of the 4 matrices ) ) ) ) i +1 0 σ 0 =, σ 1 =, σ 2 =, σ 3 = i S.69) we have ) V0 V = V µ σ µ + V 3 V 1 iv 2 = V 1 + iv 2 V 0 V 3 and hence detv ) = V 0 + V 3 )V 0 V 3 ) V 1 iv 2 )V 1 + iv 2 ) = V 2 0 V 2 3 V 2 1 V 2 2 = g αβ V α V β. S.70) The determinant of a matrix product is the product of the individual matrices determinants. Hence, for the transform S.66), detv ) = detm) detv ) detm ) = detv ) detm) 2. S.71) The M matrices of interest to us belong to the SL2, C) group they are complex matrices with units determinants. There are no other restrictions, but detm) = 1 is enough to assure detv ) = detv ), cf. eq. S.71). Thanks to the relation S.68), this means g αβ V αv β = detv ) = detv ) = g αβ V α V β S.72) which proves that the matrix L µ νm) is indeed Lorentzian. 15
16 To prove that the Lorentz transform L µ νm) is orthochronous, we need to show that for any V µ in the forward light cone V 2 > 0 and V 0 > 0 the V µ is also in the forward light cone. In matrix terms, V 2 > 0 and V 0 > 0 mean detv ) > 0 and trv ) > 0; together, these two conditions means that the 2 2 hermitian matrix V is positive-definite. The transform S.66) preserves positive definiteness: if for any complex 2-vector ξ 0 we have ξ V ξ > 0, then ξ V ξ = ξ MV M ξ = M ξ) V M ξ) > 0. S.73) Note that M ξ 0 for any ξ 0 because detm) 0.) Thus, for any M SL2, C) the Lorentz transform V µ V µ preserves the forward light cone in other words, the L µ νm) is orthochronous, L µ νm) O + 3, 1). Problem 4c ): The simplest proof the L ν µ M) is proper as well as orthochronous involves the group law part d) of this problem) and the explicit examples of a pure rotation and a pure boost in part e)), both of which are manifestly proper. For any SL2, C) matrix M we may decompose M = HU where H = MM is hermitian and U = H 1 M is unitary. Proof: UU = H 1 MM H 1 = H 1 H 2 H 1 = 1.) Furthermore, both H and U are unimodular deth) = detu) = 1), or in other words H, U SL2, C), which allows us to define two separate Lorentz transforms LH) and LU). According to the group law, together these two transform accomplish the LM) transform, LM) = LH) LU). S.74) Now, H is hermitian, unimodular, and positive definite, hence it has a well-defined logarithm which is hermitian and traceless, trlog H) = 0. For the 2 2 matrices, this means log H = 2 1rσ for some real 3 vector r, or equivalently H = exp 2 1rnσ). As we shall see in part e) below, this means that LH) is a pure Lorentz boost of rapidity r in the direction n. This boost manifestly does not invert space or time, thus LU) is proper. Likewise, U is unitary and unimodular, thus U SU2) and defines a pure rotation of space. Indeed, any U SU2) can be written as U = exp 2 i θn σ ) for some angle θ and 16
17 some axis n, and we shall see in part e) that for such U, LU) is indeed a rotation of the 3D space by angle θ around axis n. Again, this rotation is proper it does not invert space or time. Thus, LH) and LU) are both proper Lorentz transforms, hence their product LM) must also be proper. Proof: detlm)) = detlh)) detlu)) = +1.) Q.E.D. And by the way, since any proper, orthochronous Lorentz transform L SO + 1, 3) can be realized as LM) for some M SL2, C), it follows that any such transform is a product of a pure space rotation LH) followed by a pure Lorentz boost LU). Problem 4d): Let s plug L ν µ M)V ν = V µ into eqs. 17): σ µ L ν µ M)V ν = σ µ V µ = Mσ ν V ν )M ) = Mσ ν M V µ. S.75) Since this equation holds true for any 4-vector V µ, we must have σ µ L ν µ M) = Mσ ν M. S.76) This formula defines the Lorentz transform LM) in a V µ independent way, which is more convenient for verifying the group law. Indeed, for any two SL2, C) matrices M 1 and M 2, it gives us σ µ Lµ ν M 2 M 1 ) = M 2 M 1 )σ ν M 2 M 1 ) = M 2 M 1 σ ν M 1 M 2 ) = M 2 M 1 σ ν M 1 M 2 = M 2 σ ρ Lρ ν M 1 ) ) M 2 = M 2 σ ρ M 2 L ρ ν M 1 ) = σ µ Lµ ρ M 2 ) Lρ ν M 1 ) = σ µ Lµ ρ M 2 ) Lρ ν M 1 ) ) S.77) and hence L ν µ M 2 M 1 ) = L ρ µ M 2 ) L ν ρ M 1 ). S.78) In short, LM 2 M 1 ) = LM 2 )LM 1 ). Q.E.D. 17
18 Problem 4e): Let M = exp 2 i θ nσ) = cos 2 θ i sin 2 θ nσ and hence M = M 1 = cos 2 θ + i sin 2 θ nσ. Since σ 0 = 1 while the Pauli matrices σ 1,2,3 are traceless, for any unitary M Mσ 0 M = MM = 1 = σ 0 S.79) while MσM = MσM 1 = tr Mσ i M ) = tr Mσ i M 1) = tr σ i) = 0 = Mσ i M = R ij σ j for some R ij. S.80) In terms of the Lorentz transform 17), this means that V 0 = V 0 while V depends only on the space components of the V. In other words, this Lorentz transform is a rotation of space that does not affect the time. Specifically, σ V = MVσ)M = cos 2 θ + i sin 2 θ nσ) Vσ) cos 2 θ i sin 2 θ nσ) = cos 2 2 θ Vσ) i sin 2 θ cos 2 nσ, θ Vσ] = 2in V) σ ) ) + sin 2 2 θ nσ)vσ)nσ) = 2nv)nσ) Vσ) S.81) = cos θ Vσ) + sin θn V)σ) + 1 cos θ)nv)nσ), hence V = cos θ V nnv) ) + sin θ n V + nnv). S.82) This is indeed a rotation through angle θ around axis n. 18
19 Now consider the Lorentz transforms LM) for hermitian matrices M = M. Specifically, plugging M = exp 2 r nσ) = cosh 2 r sinh 2 r nσ into eq. 17) gives us σ µ V µ = V 0 σ V = MV 0 σ V)M = cosh r 2 sinh r 2 nσ) V 0 Vσ) cosh r 2 sinh r 2 nσ) = cosh 2 2 r V 0 Vσ) {nσ, sinh 2 r cosh 2 r V0 Vσ) } ) = 2V 0 nσ) 2nV) ) + sinh 2 2 r nσ)v 0 Vσ)nσ) = V 0 2nV)nσ) + Vσ) = cosh r V 0 + sinh r nv ) σn) sinh r V 0 + cosh r nv ) In other words, σ V nnv) ). S.83) V 0 = cosh r) V 0 + sinh r) nv, V = n sinh r) V 0 + cosh r) nv ) + V nnv) ), S.84) which is precisely the Lorentz boost of rapidity r in the direction n. The rapidity r is related to the usual parameters of a Lorentz boost according to β = tanh r, γ = cosh r, γβ = sinh r. For several boosts in the same directions, the rapidities add up, r tot = r 1 +r 2 +.) Q.E.D. Problem 4f): For any Lie algebra equivalent to an angular momentum or its analytic continuation, the product of two doublets comprises a triplet and a singlet, 2 2 = 3 1, or in j) notations, 1 2 ) 2 1 ) = 1) 0). Furthermore, the triplet 3 = 1) is symmetric with respect to permutations of the two doublets while the singlet 1 = 0) is antisymmetric. For two separate and independent types of angular momenta J + and J we combine the j + quantum numbers independently from the j and the j quantum numbers independently from the j +. For two bi-spinors, this gives us 2 1, 1 2 ) 1 2, 1 2 ) = 1, 1) 1, 0) 0, 1) 0, 0). S.85) Furthermore, the symmetric part of this product should be either symmetric with respect to 19
20 both the j + and the j indices or antisymmetric with respect to both indices, thus 1 2, 1 2 ) 1 2, 1 2 )] sym = 1, 1) 0, 0). S.86) Likewise, the antisymmetric part is either symmetric with respect to the j + but antisymmetric with respect to the j or the other way around, thus 1 2, 1 2 ) 1 2, 1 2 )] antisym = 1, 0) 0, 1). S.87) From the SO + 3, 1) point of view, the bi-spinor 1 2, 1 2 ) is the Lorentz vector. A general 2-index Lorentz tensor transforms like a product of two such vectors, so from the SL2, C) point of view it s a product of two bi-spinors, which decomposes to irreducible multiplets according to eq. S.85). The Lorentz symmetry respects splitting of a general 2-index tensor into a symmetric tensor T µν = +T νµ and an asymmetric tensor F µν = F νµ. The symmetric tensor corresponds to a symmetrized square of a bi-spinor, which decomposes into irreducible multiplets according to eq. S.86). The singlet 0, 0) component is the Lorentz-invariant trace T µ µ while the 1, 1) irreducible multiplet is the traceless part of the symmetric tensor. Likewise, the antisymmetric Lorentz tensor F µν eq. S.87). = F νµ decomposes according to Here, the irreducible components 1, 0) and 0, 1) are complex but conjugate to each other; individually, they describe antisymmetric tensors subject to complex duality conditions 1 2 ɛκλµν F µν = ±if κλ, i.e. E = ±ib. 20
d 3 k In the same non-relativistic normalization x k = exp(ikk),
PHY 396 K. Solutions for homework set #3. Problem 1a: The Hamiltonian 7.1 of a free relativistic particle and hence the evolution operator exp itĥ are functions of the momentum operator ˆp, so they diagonalize
More informationPHY 396 K. Solutions for problems 1 and 2 of set #5.
PHY 396 K. Solutions for problems 1 and of set #5. Problem 1a: The conjugacy relations  k,  k,, Ê k, Ê k, follow from hermiticity of the Âx and Êx quantum fields and from the third eq. 6 for the polarization
More informationPHY 396 K. Problem set #7. Due October 25, 2012 (Thursday).
PHY 396 K. Problem set #7. Due October 25, 2012 (Thursday. 1. Quantum mechanics of a fixed number of relativistic particles does not work (except as an approximation because of problems with relativistic
More informationetc., etc. Consequently, the Euler Lagrange equations for the Φ and Φ fields may be written in a manifestly covariant form as L Φ = m 2 Φ, (S.
PHY 396 K. Solutions for problem set #3. Problem 1a: Let s start with the scalar fields Φx and Φ x. Similar to the EM covariant derivatives, the non-abelian covariant derivatives may be integrated by parts
More informationE 2 = p 2 + m 2. [J i, J j ] = iɛ ijk J k
3.1. KLEIN GORDON April 17, 2015 Lecture XXXI Relativsitic Quantum Mechanics 3.1 Klein Gordon Before we get to the Dirac equation, let s consider the most straightforward derivation of a relativistically
More informationPHY 396 K. Problem set #3. Due September 29, 2011.
PHY 396 K. Problem set #3. Due September 29, 2011. 1. Quantum mechanics of a fixed number of relativistic particles may be a useful approximation for some systems, but it s inconsistent as a complete theory.
More informationProblem 1(a): Since the Hamiltonian (1) is a function of the particles momentum, the evolution operator has a simple form in momentum space,
PHY 396 K. Solutions for problem set #5. Problem 1a: Since the Hamiltonian 1 is a function of the particles momentum, the evolution operator has a simple form in momentum space, exp iĥt d 3 2π 3 e itω
More informationSpinor Formulation of Relativistic Quantum Mechanics
Chapter Spinor Formulation of Relativistic Quantum Mechanics. The Lorentz Transformation of the Dirac Bispinor We will provide in the following a new formulation of the Dirac equation in the chiral representation
More informationIntroduction to relativistic quantum mechanics
Introduction to relativistic quantum mechanics. Tensor notation In this book, we will most often use so-called natural units, which means that we have set c = and =. Furthermore, a general 4-vector will
More informationAppendix C Lorentz group and the Dirac algebra
Appendix C Lorentz group and the Dirac algebra This appendix provides a review and summary of the Lorentz group, its properties, and the properties of its infinitesimal generators. It then reviews representations
More informationParity P : x x, t t, (1.116a) Time reversal T : x x, t t. (1.116b)
4 Version of February 4, 005 CHAPTER. DIRAC EQUATION (0, 0) is a scalar. (/, 0) is a left-handed spinor. (0, /) is a right-handed spinor. (/, /) is a vector. Before discussing spinors in detail, let us
More informationLecture 19 (Nov. 15, 2017)
Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationLorentz-covariant spectrum of single-particle states and their field theory Physics 230A, Spring 2007, Hitoshi Murayama
Lorentz-covariant spectrum of single-particle states and their field theory Physics 30A, Spring 007, Hitoshi Murayama 1 Poincaré Symmetry In order to understand the number of degrees of freedom we need
More informationChapter 1 LORENTZ/POINCARE INVARIANCE. 1.1 The Lorentz Algebra
Chapter 1 LORENTZ/POINCARE INVARIANCE 1.1 The Lorentz Algebra The requirement of relativistic invariance on any fundamental physical system amounts to invariance under Lorentz Transformations. These transformations
More informationThe Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Lorentz Transformation Properties of the Dirac Field First, rotations. In ordinary quantum mechanics, ψ σ i ψ (1) is
More informationIntroduction to Modern Quantum Field Theory
Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical
More informationExercises Symmetries in Particle Physics
Exercises Symmetries in Particle Physics 1. A particle is moving in an external field. Which components of the momentum p and the angular momentum L are conserved? a) Field of an infinite homogeneous plane.
More informationThe Homogenous Lorentz Group
The Homogenous Lorentz Group Thomas Wening February 3, 216 Contents 1 Proper Lorentz Transforms 1 2 Four Vectors 2 3 Basic Properties of the Transformations 3 4 Connection to SL(2, C) 5 5 Generators of
More information1.7 Plane-wave Solutions of the Dirac Equation
0 Version of February 7, 005 CHAPTER. DIRAC EQUATION It is evident that W µ is translationally invariant, [P µ, W ν ] 0. W is a Lorentz scalar, [J µν, W ], as you will explicitly show in homework. Here
More informationRepresentations of Lorentz Group
Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the
More informationPhysics 221AB Spring 1997 Notes 36 Lorentz Transformations in Quantum Mechanics and the Covariance of the Dirac Equation
Physics 221AB Spring 1997 Notes 36 Lorentz Transformations in Quantum Mechanics and the Covariance of the Dirac Equation These notes supplement Chapter 2 of Bjorken and Drell, which concerns the covariance
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationPlan for the rest of the semester. ψ a
Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 6: Lectures 11, 12
As usual, these notes are intended for use by class participants only, and are not for circulation Week 6: Lectures, The Dirac equation and algebra March 5, 0 The Lagrange density for the Dirac equation
More informationSymmetries, Groups, and Conservation Laws
Chapter Symmetries, Groups, and Conservation Laws The dynamical properties and interactions of a system of particles and fields are derived from the principle of least action, where the action is a 4-dimensional
More informationContinuous symmetries and conserved currents
Continuous symmetries and conserved currents based on S-22 Consider a set of scalar fields, and a lagrangian density let s make an infinitesimal change: variation of the action: setting we would get equations
More informationThe Lorentz and Poincaré Groups in Relativistic Field Theory
The and s in Relativistic Field Theory Term Project Nicolás Fernández González University of California Santa Cruz June 2015 1 / 14 the Our first encounter with the group is in special relativity it composed
More informationLecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist
More informationd 2 Area i K i0 ν 0 (S.2) when the integral is taken over the whole space, hence the second eq. (1.12).
PHY 396 K. Solutions for prolem set #. Prolem 1a: Let T µν = λ K λµ ν. Regardless of the specific form of the K λµ ν φ, φ tensor, its antisymmetry with respect to its first two indices K λµ ν K µλ ν implies
More informationIntroduction to Group Theory
Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)
More informationσ 2 + π = 0 while σ satisfies a cubic equation λf 2, σ 3 +f + β = 0 the second derivatives of the potential are = λ(σ 2 f 2 )δ ij, π i π j
PHY 396 K. Solutions for problem set #4. Problem 1a: The linear sigma model has scalar potential V σ, π = λ 8 σ + π f βσ. S.1 Any local minimum of this potential satisfies and V = λ π V σ = λ σ + π f =
More informationInstructor: Sudipta Mukherji, Institute of Physics, Bhubaneswar
Chapter 1 Lorentz and Poincare Instructor: Sudipta Mukherji, Institute of Physics, Bhubaneswar 1.1 Lorentz Transformation Consider two inertial frames S and S, where S moves with a velocity v with respect
More information1. Rotations in 3D, so(3), and su(2). * version 2.0 *
1. Rotations in 3D, so(3, and su(2. * version 2.0 * Matthew Foster September 5, 2016 Contents 1.1 Rotation groups in 3D 1 1.1.1 SO(2 U(1........................................................ 1 1.1.2
More informationUnitary rotations. October 28, 2014
Unitary rotations October 8, 04 The special unitary group in dimensions It turns out that all orthogonal groups SO n), rotations in n real dimensions) may be written as special cases of rotations in a
More informationSpecial Relativity. Chapter The geometry of space-time
Chapter 1 Special Relativity In the far-reaching theory of Special Relativity of Einstein, the homogeneity and isotropy of the 3-dimensional space are generalized to include the time dimension as well.
More information7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section
More information232A Lecture Notes Representation Theory of Lorentz Group
232A Lecture Notes Representation Theory of Lorentz Group 1 Symmetries in Physics Symmetries play crucial roles in physics. Noether s theorem relates symmetries of the system to conservation laws. In quantum
More informationQuantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams
Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The
More informationH =Π Φ L= Φ Φ L. [Φ( x, t ), Φ( y, t )] = 0 = Φ( x, t ), Φ( y, t )
2.2 THE SPIN ZERO SCALAR FIELD We now turn to applying our quantization procedure to various free fields. As we will see all goes smoothly for spin zero fields but we will require some change in the CCR
More informationCHAPTER II: The QCD Lagrangian
CHAPTER II: The QCD Lagrangian.. Preparation: Gauge invariance for QED - 8 - Ã µ UA µ U i µ U U e U A µ i.5 e U µ U U Consider electrons represented by Dirac field ψx. Gauge transformation: Gauge field
More informationPhysics 557 Lecture 5
Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as
More informationProblem 1(a): As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as L (D µ φ) L
PHY 396 K. Solutions for problem set #. Problem a: As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as D µ D µ φ φ = 0. S. In particularly,
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationL = e i `J` i `K` D (1/2,0) (, )=e z /2 (10.253)
44 Group Theory The matrix D (/,) that represents the Lorentz transformation (.4) L = e i `J` i `K` (.5) is D (/,) (, )=exp( i / /). (.5) And so the generic D (/,) matrix is D (/,) (, )=e z / (.53) with
More informationQuantization of the Spins
Chapter 5 Quantization of the Spins As pointed out already in chapter 3, the external degrees of freedom, position and momentum, of an ensemble of identical atoms is described by the Scödinger field operator.
More informationFunctional determinants
Functional determinants based on S-53 We are going to discuss situations where a functional determinant depends on some other field and so it cannot be absorbed into the overall normalization of the path
More informationQuantum Field Theory III
Quantum Field Theory III Prof. Erick Weinberg January 19, 2011 1 Lecture 1 1.1 Structure We will start with a bit of group theory, and we will talk about spontaneous symmetry broken. Then we will talk
More informationPHY 396 K. Solutions for homework set #9.
PHY 396 K. Solutions for homework set #9. Problem 2(a): The γ 0 matrix commutes with itself but anticommutes with the space-indexed γ 1,2,3. At the same time, the parity reflects the space coordinates
More informationProblem 1(a): The scalar potential part of the linear sigma model s Lagrangian (1) is. 8 i φ2 i f 2) 2 βλf 2 φ N+1,
PHY 396 K. Solutions for problem set #10. Problem 1a): The scalar potential part of the linear sigma model s Lagrangian 1) is Vφ) = λ 8 i φ i f ) βλf φ N+1, S.1) where the last term explicitly breaks the
More informationREVIEW. Quantum electrodynamics (QED) Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field:
Quantum electrodynamics (QED) based on S-58 Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field: Noether current of the lagrangian for a free Dirac
More informationMaxwell s equations. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationx 3 x 1 ix 2 x 1 + ix 2 x 3
Peeter Joot peeterjoot@pm.me PHY2403H Quantum Field Theory. Lecture 19: Pauli matrices, Weyl spinors, SL2,c, Weyl action, Weyl equation, Dirac matrix, Dirac action, Dirac Lagrangian. Taught by Prof. Erich
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationPHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain
PHY 396 K. Solutions for problem set #6. Problem 1(a): Starting with eq. (3) proved in class and applying the Leibniz rule, we obtain γ κ γ λ, S µν] = γ κ γ λ, S µν] + γ κ, S µν] γ λ = γ κ( ig λµ γ ν ig
More informationAngular Momentum set II
Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. Non-Relativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e - e + - + e - e + - + e - e + - + e - e + - Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 Non-Relativistic QM (Revision)
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More informationLecture 5: Sept. 19, 2013 First Applications of Noether s Theorem. 1 Translation Invariance. Last Latexed: September 18, 2013 at 14:24 1
Last Latexed: September 18, 2013 at 14:24 1 Lecture 5: Sept. 19, 2013 First Applications of Noether s Theorem Copyright c 2005 by Joel A. Shapiro Now it is time to use the very powerful though abstract
More information1 Free real scalar field
1 Free real scalar field The Hamiltonian is H = d 3 xh = 1 d 3 x p(x) +( φ) + m φ Let us expand both φ and p in Fourier series: d 3 p φ(t, x) = ω(p) φ(t, x)e ip x, p(t, x) = ω(p) p(t, x)eip x. where ω(p)
More informationLorentz Group and Lorentz Invariance
Chapter 1 Lorentz Group and Lorentz Invariance In studying Lorentz-invariant wave equations, it is essential that we put our understanding of the Lorentz group on firm ground. We first define the Lorentz
More information2. Lie groups as manifolds. SU(2) and the three-sphere. * version 1.4 *
. Lie groups as manifolds. SU() and the three-sphere. * version 1.4 * Matthew Foster September 1, 017 Contents.1 The Haar measure 1. The group manifold for SU(): S 3 3.3 Left- and right- group translations
More information11 Spinor solutions and CPT
11 Spinor solutions and CPT 184 In the previous chapter, we cataloged the irreducible representations of the Lorentz group O(1, 3. We found that in addition to the obvious tensor representations, φ, A
More informationPhysics 218 Polarization sum for massless spin-one particles Winter 2016
Physics 18 Polarization sum for massless spin-one particles Winter 016 We first consider a massless spin-1 particle moving in the z-direction with four-momentum k µ = E(1; 0, 0, 1). The textbook expressions
More informationRepresentations of the Poincaré Group
Representations of the Poincaré Group Ave Khamseh Supervisor: Professor Luigi Del Debbio, University of Edinburgh Last updated: July 28, 23 Contents The Poincaré group. Characterisation of the Poincaré
More informationParticle Physics Dr. Alexander Mitov Handout 2 : The Dirac Equation
Dr. A. Mitov Particle Physics 45 Particle Physics Dr. Alexander Mitov µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - µ + e - e + µ - Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 46 Non-Relativistic
More information2 Quantization of the scalar field
22 Quantum field theory 2 Quantization of the scalar field Commutator relations. The strategy to quantize a classical field theory is to interpret the fields Φ(x) and Π(x) = Φ(x) as operators which satisfy
More informationA Brief Introduction to Relativistic Quantum Mechanics
A Brief Introduction to Relativistic Quantum Mechanics Hsin-Chia Cheng, U.C. Davis 1 Introduction In Physics 215AB, you learned non-relativistic quantum mechanics, e.g., Schrödinger equation, E = p2 2m
More information1.3 Translational Invariance
1.3. TRANSLATIONAL INVARIANCE 7 Version of January 28, 2005 Thus the required rotational invariance statement is verified: [J, H] = [L + 1 Σ, H] = iα p iα p = 0. (1.49) 2 1.3 Translational Invariance One
More information2.4 Parity transformation
2.4 Parity transformation An extremely simple group is one that has only two elements: {e, P }. Obviously, P 1 = P, so P 2 = e, with e represented by the unit n n matrix in an n- dimensional representation.
More informationLectures April 29, May
Lectures 25-26 April 29, May 4 2010 Electromagnetism controls most of physics from the atomic to the planetary scale, we have spent nearly a year exploring the concrete consequences of Maxwell s equations
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More information129 Lecture Notes More on Dirac Equation
19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large
More informationTextbook Problem 4.2: We begin by developing Feynman rules for the theory at hand. The Hamiltonian clearly decomposes into Ĥ = Ĥ0 + ˆV where
PHY 396 K. Solutions for problem set #11. Textbook Problem 4.2: We begin by developing Feynman rules for the theory at hand. The Hamiltonian clearly decomposes into Ĥ = Ĥ0 + ˆV where Ĥ 0 = Ĥfree Φ + Ĥfree
More informationMSci EXAMINATION. Date: XX th May, Time: 14:30-17:00
MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question
More informationMandl and Shaw reading assignments
Mandl and Shaw reading assignments Chapter 2 Lagrangian Field Theory 2.1 Relativistic notation 2.2 Classical Lagrangian field theory 2.3 Quantized Lagrangian field theory 2.4 Symmetries and conservation
More informationCHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS
CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS 1.1 PARTICLES AND FIELDS The two great structures of theoretical physics, the theory of special relativity and quantum mechanics, have been combined
More informationShow, for infinitesimal variations of nonabelian Yang Mills gauge fields:
Problem. Palatini Identity Show, for infinitesimal variations of nonabelian Yang Mills gauge fields: δf i µν = D µ δa i ν D ν δa i µ..) Begin by considering the following form of the field strength tensor
More informationPath Integral for Spin
Path Integral for Spin Altland-Simons have a good discussion in 3.3 Applications of the Feynman Path Integral to the quantization of spin, which is defined by the commutation relations [Ŝj, Ŝk = iɛ jk
More informationProblem 1(a): As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as L (D µ φ) L
PHY 396 K. Solutions for problem set #. Problem 1a: As discussed in class, Euler Lagrange equations for charged fields can be written in a manifestly covariant form as D µ D µ φ φ = 0. S.1 In particularly,
More informationThe Lorentz and Poincaré groups. By Joel Oredsson
The Lorentz and Poincaré groups By Joel Oredsson The Principle of Special Rela=vity: The laws of nature should be covariant with respect to the transforma=ons between iner=al reference frames. x µ x' µ
More informationMathematical Methods wk 2: Linear Operators
John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationLorentz Invariance and Second Quantization
Lorentz Invariance and Second Quantization By treating electromagnetic modes in a cavity as a simple harmonic oscillator, with the oscillator level corresponding to the number of photons in the system
More information3 Quantization of the Dirac equation
3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary
More informationChapter 17 The bilinear covariant fields of the Dirac electron. from my book: Understanding Relativistic Quantum Field Theory.
Chapter 17 The bilinear covariant fields of the Dirac electron from my book: Understanding Relativistic Quantum Field Theory Hans de Vries November 10, 008 Chapter Contents 17 The bilinear covariant fields
More informationbe stationary under variations in A, we obtain Maxwell s equations in the form ν J ν = 0. (7.5)
Chapter 7 A Synopsis of QED We will here sketch the outlines of quantum electrodynamics, the theory of electrons and photons, and indicate how a calculation of an important physical quantity can be carried
More informationMaxwell s equations. based on S-54. electric field charge density. current density
Maxwell s equations based on S-54 Our next task is to find a quantum field theory description of spin-1 particles, e.g. photons. Classical electrodynamics is governed by Maxwell s equations: electric field
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14.
As usual, these notes are intended for use by class participants only, and are not for circulation. Week 7: Lectures 13, 14 Majorana spinors March 15, 2012 So far, we have only considered massless, two-component
More informationQuantum Field Theory Example Sheet 4 Michelmas Term 2011
Quantum Field Theory Example Sheet 4 Michelmas Term 0 Solutions by: Johannes Hofmann Laurence Perreault Levasseur Dave M. Morris Marcel Schmittfull jbh38@cam.ac.uk L.Perreault-Levasseur@damtp.cam.ac.uk
More informationLecture notes for Mathematical Physics. Joseph A. Minahan 1 Department of Physics and Astronomy Box 516, SE Uppsala, Sweden
Lecture notes for Mathematical Physics Joseph A. Minahan 1 Department of Physics and Astronomy Box 516, SE-751 20 Uppsala, Sweden 1 E-mail: joseph.minahan@fysast.uu.se 1 1 Introduction This is a course
More informationiδ jk q 2 m 2 +i0. φ j φ j) 2 φ φ = j
PHY 396 K. Solutions for problem set #8. Problem : The Feynman propagators of a theory follow from the free part of its Lagrangian. For the problem at hand, we have N scalar fields φ i (x of similar mass
More informationADVANCED TOPICS IN THEORETICAL PHYSICS II Tutorial problem set 2, (20 points in total) Problems are due at Monday,
ADVANCED TOPICS IN THEORETICAL PHYSICS II Tutorial problem set, 15.09.014. (0 points in total) Problems are due at Monday,.09.014. PROBLEM 4 Entropy of coupled oscillators. Consider two coupled simple
More informationPHY 396 K. Problem set #11, the last set this semester! Due December 1, 2016.
PHY 396 K. Problem set #11, the last set this semester! Due December 1, 2016. In my notations, the A µ and their components A a µ are the canonically normalized vector fields, while the A µ = ga µ and
More informationRotational motion of a rigid body spinning around a rotational axis ˆn;
Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with
More informationLecture 4 - Relativistic wave equations. Relativistic wave equations must satisfy several general postulates. These are;
Lecture 4 - Relativistic wave equations Postulates Relativistic wave equations must satisfy several general postulates. These are;. The equation is developed for a field amplitude function, ψ 2. The normal
More informationQUALIFYING EXAMINATION, Part 2. Solutions. Problem 1: Quantum Mechanics I
QUALIFYING EXAMINATION, Part Solutions Problem 1: Quantum Mechanics I (a) We may decompose the Hamiltonian into two parts: H = H 1 + H, ( ) where H j = 1 m p j + 1 mω x j = ω a j a j + 1/ with eigenenergies
More informationSymmetries, Fields and Particles 2013 Solutions
Symmetries, Fields and Particles 013 Solutions Yichen Shi Easter 014 1. (a) Define the groups SU() and SO(3), and find their Lie algebras. Show that these Lie algebras, including their bracket structure,
More informationmsqm 2011/8/14 21:35 page 189 #197
msqm 2011/8/14 21:35 page 189 #197 Bibliography Dirac, P. A. M., The Principles of Quantum Mechanics, 4th Edition, (Oxford University Press, London, 1958). Feynman, R. P. and A. P. Hibbs, Quantum Mechanics
More informationFrom Particles to Fields
From Particles to Fields Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 25, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, 2011 1 / 24 Hamiltonian
More information