(a) As a reminder, the classical definition of angular momentum is: l = r p

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1 PHYSICS T8: Standard Model Midter Exa Solution Key (216) 1. [2 points] Short Answer ( points each) (a) As a reinder, the classical definition of angular oentu is: l r p Based on this, what are the units of angular oentu in natural units? [l] [E] You can get this directly by noting that spin has units of angular oentu in ters of h and h 1. But also [r] [E] 1 and [p] [E] 1. (b) Let s do a short essay question (1 or 2 sentences). How does the resolution of the negative energy positrons in classical Dirac theory relate to the Pauli Exclusion Principle? There s a lot of flexibility in this answer, but I a essentially looking for three things: 1) Treated as a classical theory, the positrons have a negative energy, but the sae charge as the electron. 2) Quantization of ferions involves postuating an ANTI-coutation relation for creation and annihilation operators, which reverses the charge and energy for the positron. ) That negative sign eans that exchanges of particles are negatives of one another. Hence, p, p p, p for ferions, which is not possible unless the state is the vacuu. (c) Please give a short description of each of the big three discrete transforations, C, P, and T. C: Charge conjugation; reverse all electrical (and other quantu) charges P: Parity; reverse spatial coordinates T: Tie inverstion; reverse the arrow of tie. (d) I d like you to construct the siplest possible finite group that includes i under ordinary ultiplication. Given the requireents for a group, list all possible ebers. E.C. Is this group Abellian? The group is Abellian, since order clearly doesn t atter. The eleents and ultiplication table are: 1-1 i -i i -i i i i i -i i -i i [2 points] Consider a particle with the -oentu: p µ GeV (a) What is the ass of the particle? Based on the approxiate answer (within 1%), what type(s) of particle ight this be?

2 The ass is siply: 2 E 2 p 2 ( ) GeV 2 1 GeV 2 16 This is a proton or a neutron. 1 GeV (b) Now boost the particle by v.8 in the z-direction (the speed should increase). What is the -oentu in the boosted frae? Call it p µ. First note: γ ; vγ So the Boost is: p µ 1 1 GeV GeV (c) What is the speed of the particle in the pried (boosted) frae? We calculate the speed via: so (d) Take a second vector: v p p v 7.96 A µ as easured in the unboosted frae. What is p A as easured in the boosted frae? Tricked you! (Or hopefully not.) The dot product doesn t depend on frae so it s way easier to do it in the unbarred frae: A p 8 ( 8 ) ( ) 1

3 . [2 points] Consider our old friend, the coplex scalar field, but with an additional quartic potential: L µ φ µ φ 2 φφ 1 2 λ(φφ ) 2 (a) Copute the Euler Lagrange equation for φ. First note: ( µ φ ) µ φ and so: φ 2 φ φ 2 φ µ µ φ 2 φ φ 2 φ (b) Copute the T 11 coponent of the stress-energy tensor. First, note: T 11 ( 1 φ a ) 1 φ a + L where I ve used the fact that g Expanding out: T φ 1 φ + L φ φ φ φ φ 1 φ V (φ) φ φ 2 φ 2 φ φ φ + 1 φ 1 φ V (φ) (c) For an isotropic field, T 11 corresponds to the pressure. Isotropy eans that: 1 φ 1 φ 2 φ 2 φ φ φ 1 φ φ What is the pressure in the isotropic liit? Note: Your answer shouldn t include individual spatial derivatives, only gradients. In that case: P φ φ 1 φ φ V (φ) (d) The Lagrangian is invariant under a U(1) syetery transforation: φ e iqθ φ ; φ e +iqθ φ What is the Noether current for this syetry for our coplex field? In soe sense this is a trick, since the answer is identical to what we get for a coplex field with any other potential. First note: φ θ iqφ and φ θ iqφ and ( µ φ ) µ φ with a siilar expression (coplex conjugate) for φ. Cobining: J µ iq(φ µ φ φ µ φ )

4 (e) We have seen that a coplex scalar field ay be decoposed into plane waves: φ(x) d p 1 ( bp (2π) 2 e ip x + c pe ip x) 2Ep with a quantization of the for: ˆφ (...)(ˆb + ĉ ) where ˆb annihilates a φ particle, and ĉ creates a φ particle, and a transposed version for ˆφ. Iagine expanding out the quartic potential (interaction) ter in the Lagrangian: ˆL Int 1 2 ˆφ ˆφ ˆφ ˆφ In addition to the prefactors and explicit integrals (which you should ignore), the quantized version of the interaction potential will have cobinations of creation and annihilation ters. Write at least cobinations of creation/annihilation ters arising fro quantizing the field. Assuing φ particles have a charge of +1 and φ have a charge of 1, what would be the net charge of these reactions? Note: It sees like I asking a lot, but I just want quartets of possible operator cobinations. First note the general expansion: which yields: +2 φ-2φ: No change in charge ˆL int λ(ˆb + ĉ )(ˆb + ĉ )(ˆb + ĉ)(ˆb + ĉ) λˆbˆbˆb ˆb +2φ-1φ+φ : No change in charge λˆbˆbˆb ĉ +1φ-1φ+1φ -1φ : No change in charge ˆbĉ ĉˆb

5 . [2 points] We have only occasionally encountered the group SO(), a group which is isoorphic to the ore failiar SU(2). Like SU(2), SO() has three generators: i i X 1 i ; X 2 ; X i i i (a) What is the coutator, [X 1, X 2 ]? If, indeed, our generators for a coplete set, your answer should be a superposition of the generators above. [X 1, X 2 ] 1 1 ix (b) Copute the generalized for of M 1 (θ), where the rotation is around the x-axis (X 1 ). You ay find the trigonoetric expansions helpful: cos x 1 x2 2 + x!... We ve done this before. We get: sin x x x 6 + x!... M 1 1 cos θ sin θ sin θ cos θ (c) Show that the atrix eleent, M 1, is orthogonal and special. This is intended to be siply a sanity check. The deterinant is: so it s Special. As for orthogonality: 1 cos θ cos θ (sin θ( sin θ) 1) 1 M 1 M T the lower 2x2 can be treated as its own atrix. The transpose siply reverses the sign on the sin, which is, of course, the inverse. (d) The point of syetries in field theories is that they leave Lagrangians invariant under transforations, including ters like Φ T Φ. Consider a specific state for a triplet of real-valued scalar fields: Φ 1 1 For the specic case of a particle at rest: What is the Q 1 of the state above? Q 1 iqφ T X 1 Φ.

6 We begin with so XΦ i i Q 1 iqφ T XΦ 2q. [1 points] Consider a specific positive-energy, spin-up solution to the Dirac equation: 1 u + E + p E+p e ip x where in this case, p refers specifically to the z-oentu of the particle. (a) Copute and siplify u + γ u +. First, note: so uγ u N 2 ( 1 γ γ E+p ) [ (E ) 2 + p N 2 1] 2 [ E 2 + p 2 + 2pE 2 ] E + p 2 1 [ 2p 2 + 2pE ] E + p 2p E+p (b) (1 points) We found that the Hailtonian of a state is: Ĥ γ ( iγ i i + ) Copute Ĥu +. You ay find the process involved in the last part helpful. First, note: u + ip u + ipu + where the derivative is pulled down fro the exponential.

7 So: Ĥu + γ ( iγ + )u + (pγ + I)u + p γ p p p + p(e+p) γ p + E + p E 2 +Ep+p 2 E E 2 +Ep 1 E+p Eu + as expected, where I ignored the noralization and the exponential. (c) E.C.: points In class I noted that the probability of easuring a particle as right-handed versus left-handed is: We have: ψ 2 P R ψ1 2 + ψ2 Recalling that p/e v, copute the probability as a function of v. Siplify as uch as possible. P R (E+p) 2 2 (E+p) E 2 + 2p + p 2 E 2 + 2pE + p E2 + 2p + p 2 2E 2 + 2pE 1 + 2v + v2 2(1 + v) (1 + v)2 2(1 + v) 1 + v 2 Cool. P R (v 1) at P R (v )., and P R (v 1) 1.