Representation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
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1 Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac group. Defnton A (fnte-dmensonal complex) representaton of G s a fnte-dmensonal vector space V over C together wth a smooth group homomorphsm G GL(V ). Equvalently, a representaton of G on V s a homomorphsm G GL(V ) such that the acton map G V V s smooth. Yet another equvalent defnton s an acton of G on V such that G V V s smooth and g : V V s C-lnear for all g G. A morphsm of G-representatons ϕ : V W s a C-lnear map satsfyng ϕ(g v) = g ϕ(v) for all v V. Sometmes such a morphsm s called an ntertwnng operator. The homomorphsm SO 2 (R) C = GL 1 (C) gven by [ ] cos θ sn θ e θ sn θ cos θ defnes a one-dmensonal complex representaton of the crcle group SO 2 (R). Ths homomorphsm s njectve wth mage the untary group U 1. Another example of a representaton s the tautologcal acton of SU 2 GL 2 (C) on C 2, sometmes called thefundamental representaton of SU 2. The group SO 3 (R) also has a fundamental representaton of sorts. Namely, the tautologcal acton of SO 3 (R) GL 3 (R) on R 3,.e. the acton by rotatons, extends to a lnear acton on C 3 n the natural way. 1.2 Let V be a representaton of G. A subspace W V s called G-stable f for any w W, we have g w W. In ths case one says that W s a subrepresentaton of V. Defnton A fnte-dmensonal representaton V of a group G s called rreducble f the only subrepresentatons of V are 0 and V tself. Obvously, any one-dmensonal representaton s rreducble. The tautologcal 2-dmensonal representaton of SU 2 s rreducble: f t were not there would exst an SU 2 -stable one-dmensonal subspace L C 2. Ths s absurd, as t s not hard to see that SU 2 acts transtvely on the set of one-dmensonal subspaces of C 2, and n partcular fxes none of them. Exercse Show that the 3-dmensonal representaton of SO 3 (R) constructed above s rreducble. 1
2 On the other hand, the fundamental representaton of SO 2 (R) GL 2 (R) GL 2 (C) on C 2 s not rreducble. Namely, the one-dmensonal subspaces spanned by (1, ) and (1, ) are SO 2 (R)-stable, snce [ ) ) cos θ sn θ ] ( 1 sn θ cos θ = e θ ( 1 and [ ] ( cos θ sn θ 1 sn θ cos θ ) ( = e θ 1 ). Exercse Prove that a morphsm between rreducble representatons s ether zero or an somorphsm. Hnt: the kernel and mage of a morphsm of G-representatons are G-stable. Wth a lttle addtonal work, we obtan the followng. Lemma (Schur). If V s a fnte-dmensonal rreducble representaton of G, then the only G- endomorphsms of V are the scalars C. Proof. Suppose A : V V s a lnear operator whch commutes wth the acton of G. Snce V s a fntedmensonal complex vector space, the operator A has an egenvalue λ C. Ths means that equaton Av = λv has nonzero solutons,.e. the operator A λi has nonzero kernel. But then Exercse mples that A λi = 0,.e. A s the scalar operator correspondng to λ. An mmedate consequence of ths s the followng. Proposton If G s abelan, then any fnte-dmensonal rreducble representaton of G s onedmensonal. Proof. Snce elements of G commute amongst each other, the mage of the homomorphsm G GL(V ) conssts of G-morphsms. By Lemma 1.2.4, ths mples that the mage s contaned n C,.e. G acts by scalars. Thus any lnear subspace of V s G-stable, but snce V s rreducble ths s only possble f V s one-dmensonal. A basc example of ths s the followng. Exercse Prove that the rreducble representatons of U 1 = SO2 (R) all have the form e θ e nθ for some n Z (as homomorphsms U 1 C = GL 1 (C)). For example, we saw n the prevous secton that the standard representaton of SO 2 (R) on C 2 decomposes as the sum of the one-dmensonal representatons correspondng to 1 and Now we ntroduce nner products. Defnton A untary representaton of G s a representaton on a Hlbert space H such that the nner product s G-nvarant, meanng g v, g w = v, w for all g G and v, w H. A slghtly more flexble noton s the followng. We say that a complex representaton V of G s untarzable f V admts a G-nvarant Hermtan nner product. Choosng such an nner product and completng V f necessary, one obtans a untary representaton. To state the key consequence of untarzablty, we ntroduce the followng termnology. A representaton V of G s called completely decomposable f there exsts a drect sum decomposton V = V 2
3 such that each V s an rreducble subrepresentaton of V. For example, any rreducble representaton s completely decomposable n a trval fashon. The prevously dscussed standard representaton of SO 2 (R) on C 2 s a non-rreducble example. For a typcal example of a representaton whch s not completely decomposable, let G = R under addton and consder the representaton R GL 2 (C) gven by [ ] 1 a a. 0 1 Proposton Let V be a untarzable representaton of G. Then V s completely decomposable. Proof. It suffces to show that any G-subrepresentaton W V admts a G-stable complement, meanng V = W W for some subrepresentaton W V. Choose a G-nvarant nner product The G-nvarance of the nner product mples that s G-stable, and snce V = W W we are done., : V V C. W := {v V v, w = 0} Fnally, we have the followng basc result on compact groups. Proposton If G s compact, then any representaton of G s untarzable. Proof. Let V be a representaton of G. Choose an nner product, : V V C arbtrarly. For example, one can choose a bass {v } of V, and then put a v, b v := a b. We defne a new nner product by the formula v, w G := g v, g w. The ntegral s well-defned because G s compact, and ths nner product s G-nvarant because g v, g w G = hg v, hg w = hg v, hg w = v, w G. g G h G hg G Corollary If G s compact, then any representaton of G s completely decomposable. 2 Le algebra representatons and quantum mechancs 2.1 Let g be a Le algebra over R. Defnton A (complex) representaton of g s a complex vector space V together wth a homomorphsm g gl(v ) of Le algebras over R. 3
4 Here the commutator bracket on gl(v ) := End C (V ) makes t a Le algebra over C, and we can then restrct scalars and vew t as a Le algebra over R. Alternatvely, one can extend scalars and construct the vector space g R C = g g over C, whch has a unque C-blnear Le bracket whch restrcts to the gven R-blnear bracket on g. The Le algebra g R C over C s called the complexfcaton of g. A representaton of g on V s equvalently a homomorphsm g R C gl(v ) of Le algebras over C. A morphsm of g-representatons ϕ : V W s a C-lnear map whch ntertwnes the g-actons: x ϕ(v) = ϕ(x v) for all x g and v V. Other notons relatng to representatons of groups, such as rreducblty, complete decomposablty, etc., are easly modfed to the case of Le algebras. Now suppose g s the Le algebra attached to the real algebrac group G. Then a group representaton ρ : G GL(V ) gves rse to a Le algebra representaton ρ : g gl(v ) by dfferenton, as follows. Gven x g, one can choose a path γ : ( ɛ, ɛ) G such that γ(0) = I and γ (0) = x. Then we defne ρ (x) := d ( ρ(γ(t))). dt For example, the tautologcal representaton of SU 2 on C 2 dfferentates to the tautologcal representaton of su 2 on C 2. We clam that the correspondng homomorphsm t=0 ρ : su 2 R C gl 2 (C) s an somorphsm onto sl 2 (C). Snce su 2 sl 2 (R), the mage of ρ s contaned n sl 2 (C). But ρ s clearly njectve, and snce su 2 R C and sl 2 (C) are both 3-dmensonal over C the clam follows. More explctly, recall that σ 1, σ 2, σ 3 s an R-bass of su 2, where σ 1, σ 2, σ 3 are the Paul matrces. The standard C-bass of sl 2 (C) s e = [ ] 0 1, f = 0 0 [ ] [ ] , h = Then we have e = 1 2 (σ 1 σ 2 ), f = 1 2 (σ 1 + σ 2 ), h = σ An observable quantty, correspondng to a self-adjont operator A : H H, s called a conserved quantty f ts egenstates are steady states. In other words, f at some tme the system s n a state such that the observable A has a defnte value, then the system wll reman n that state for all tme. Proposton An observable A : H H s conserved f and only f t commutes wth the Hamltonan operator H,.e. [H, A] = 0. Proof. Accordng the Schrödnger equaton dψ dt = Hψ, beng a steady state s equvalent to beng an egenstate of H. On the other hand, by basc lnear algebra the dagonalzable lnear operators H and A commute f and only f they are smultaneously dagonalzable,.e. any egenstate of one s an egenstate of the other. 4
5 Now we begn to dscuss the role of symmetry n quantum mechancs. Suppose that H s the state space of a quantum system. Then we nterpret a structure of untary G-representaton on H as symmetres of the physcal system, provded that the acton of G commutes wth the Hamltonan operator H,.e. H(g v) = g H(v) for all g G and v V. Proposton If H s a untary representaton of G, then g acts on H by skew-hermtan operators. Proof. Choosng an orthonormal bass C n = H, the untarty condton says that the mage of the acton homomorphsm ρ : G GL n (C) s contaned n U n. Thus the mage of ρ : g gl n (C) s contaned n the Le algebra u n of U n, whch we have seen conssts of skew-hermtan operators. More concsely: dfferentatng the equaton for g G and v, w V yelds the equaton for x g. g v, g w = v, w x v, w + v, x w = 0 In partcular, any element x g gves rse to a self-adjont operator ρ (x) on H. The followng s the quantum-mechancal analogue of Nother s theorem. Proposton If the acton of G commutes wth the Hamltonan, then for any x g, the observable ρ (x) s a conserved quantty. Proof. It s not hard to show that f the acton of G commutes wth H, so does the nduced acton of g. The clam follows mmedately from Proposton For example, let H = C 2 be the state space of a fxed spn 1 2 partcle n a magnetc feld B = (0, 0, B 3). Then H = B 3 σ 3 up to a postve scalar, so the observable A = 1 2 σ 3 correspondng to measurement of spn along the z-axs s conserved, as we have seen. On the other hand, f A su 2 s a self-adjont operator on C 2 whch s not proportonal to σ 3, then [A, H] 0 and the observable quantty correspondng to A s not conserved. 5
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