Lecture notes Particle Physics II Quantum Chromo Dynamics 1. SU(2) and SU(3) Symmetry Michiel Botje Nikhef, Science Park, Amsterdam December 7, 2012 Symmetry in (particle) physics If the Lagrangian of the World would be fully known we could • derive the equations of motion from it, and the symmetries of Nature and the conservation laws would automatically follow. For instance the Maxwell Lagrangian yields, via the Maxwell equa- • tions, all the symmetries and conservation laws of electrodynamics. In subatomic physics the Lagrangians are not so obvious, and sym- • metry considerations provide essential clues to construct them. It can be shown that an invariance of the Lagrangian under a • symmetry operation leads to a conserved quantity (Noether's the- orem). Thus, if a symmetry is found, the hunt is open for the related conservation law, and if a conservation law is found, the hunt is open for the related symmetry. For instance we know that electric charge is conserved in all reactions of elementary particles, but what symmetry is responsible for this charge conservation?11 As will become clear later, it turns out that continuous symme- • tries lead to additive conserved quantum numbers (e.g. rotation invariance angular momentum conservation) while discrete sym- ! metries lead to multiplicative conserved quantum numbers (e.g. reflection symmetry parity conservation). ! We will now use some elementary non-relativistic quantum me- • chanics to establish the relation between symmetries and constants of motion. 11The answer will be given in the next lecture. 1{3 When is an observable conserved? The expectation value of a quantum mechanical operator F is • F F with Hermitian conjugate F ∗ F y h i ≡ h j j i h i ≡ h j j i The expectation value of an observable should be a real number • so that F = F ∗. It immediately follows that the quantum h i h i mechanical operator of an observable should be Hermitian F = F y if F is observable h i Because energy is an observable it follows that the Hamiltonian • energy operator H is Hermitian. Thus we have for the Schr¨odinger equation and its Hermitian conjugate @ @ i j i = H i h j = H @t j i − @t h j This immediately leads to • @ F h i = i HF FH = 0 HF FH = 0 @t h j − j i , − Thus F is a constant of motion when F commutes with H • h i [F; H] = FH HF = 0 − A theorem of quantum mechanics states that two operators can be • simultaneously diagonalised if (and only if) they commute. Thus, if F is a constant of motion, the eigentates of H can be h i chosen such that they are also eigenstates of F H = E j i j i F = f j i j i 1{4 How do we find a conserved observable? Obviously when the Hamiltonian (or Lagrangian) is known, we • can search for commuting Hermitian operators whose expectation values are then constants of motion. However, when H is not fully known, it turns out that it is sufficient to establish (or postulate) the invariance of H under a symmetry operation. To see this, consider a transformation operator U that trans- • forms one wave function into another 0 = U j i j i Wave functions are always normalized so that we must have • 0 0 = U yU = 1 h j i h j j i It follows that the transformation operator must be unitary • U yU = UU y = I We say that U is a symmetry operator when 0 obeys the • j i same Schr¨odingerequation as . Then (U time independent) j i @U @ 1 i j i = HU i j i = U − HU = H @t j i ! @t j i Iwant j i and thus 1 U − HU = H or [H; U] = 0 Thus a symmetry operator U commutes with the Hamiltonian, • just as our conserved operator F on page 1{4. But U is unitary, and not necessarily Hermitian so that we cannot per se identify U with an observable constant of motion. 1{5 Hermitian, unitary, or both We have introduced quantum mechanical operators F which, when • Hermitian, correspond to observable expectation values F . Also, h i when F commutes with H then F is a constant of motion. h i We have also introduced transformation operators U which trans- • form one wave function into another. This operator is unitary be- cause normalisation must be preserved. When U commutes with the Hamiltonian, it leaves the Schr¨odingerequation invariant. Such unitary commuting operators are called symmetry operators. Our findings up to now are best summarised in a table • Operator Unitary Hermitian Commuting Observable x yes x Conserved observable x yes yes Transformation yes x x Symmetry transformation yes x yes Conserved + symmetry yes yes yes Thus to link a symmetry to a constant of motion, we can first try • to look for operators that are both unitary and Hermitian (and commute with the Hamiltonian). 1{6 Hermitian unitary transformations In our search for Hermitian unitary transformations, let us take a • unitary operator for which U 2 = I Multiplying from the right with U y and using UU y = I we find that U = U y so that the operator is both unitary and Hermitian. Examples are the charge conjugation operator C that turns parti- • cles into antiparticles (and vice versa) and the parity operator P that reflects the space coordinates (r r).12 ! − We can thus immediately say that if the Hamiltonian is invariant • under C or P , then the observables related to these operators are conserved quantities. Are these the only possible unitary Hermitian operators? The • answer is no, and an example is given by the Pauli spin matrices which are also both unitary and Hermitian 0 1 0 i 1 0 σx = ; σy = − ; σz = 1 0 i 0 0 1 − Let U1 and U2 be two discrete symmetry operators with • U1 = λ1 ;U2 = λ2 and U1U2 = λ1λ2 j i j i j i j i j i j i The (conserved) quantum numbers are obviously multiplicative. In these lectures we are not so much interested in discrete trans- • formations (like C, P , T ) but, instead, in continuous transforma- tions. These transformations are unitary (by definition), but not necessarily Hermitian. But the generator of a unitary continuous transformation is Hermitian, as we will see. 12The time reversal operator T also has T 2 = I but it is antiunitary, and not unitary. 1{7 Continuous transformations There is a large class of continuous transformations that • depend on one or more continuous parameters, say α 0 = U(α) j i j i An example is the transformation induced by a rotation of the wave function, or the coordinate system, over an angle α. Such transformations have the property that they can be written • as a succession of infinitesimal deviations from the identity iα n U(α) = lim I + F = exp(iαF ) n n !1 The factor `i' is a matter of definition but important (see below). In the above, F is called the generator of U. Exponentiation of an operator F should be interpreted as • 1 exp(iαF ) = I + iαF + (iαF )2 + 2! ··· Watch out: eAeB = eA+B is only true when A and B commute. Now if U is unitary (U yU = I) we have, to first order in α, • U yU = (I iαF y)(I + iαF ) = I + iα(F F y) = I − − so that F = F y. In other words, the generator of a unitary oper- ator is Hermitian. Now we also understand the factor `i' in the definition of a genera- • tor: without it the generator G iF of a unitary operator would ≡ not be Hermitian but anti-Hermitian: G = Gy − 1{8 Generators as conserved observables We have seen that if U is a symmetry operator, then it is unitary • and commutes with the Hamiltonian. The generator F of U is Hermitian and will also commute with H, as we will now show. First, if U(α) is a symmetry operator then the infinitesimal trans- • formation U() will also be a symmetry operator. Expanding to the first order in obtains [H; U] =: [H; I + iF ] = [H; I] +i [H; F ] = 0 [H; F ] = 0 ! 0 The generator F of a symmetry| {z } transformation is thus both Her- • mitian and commutes with H so that its expectation value is an observable constant of motion. Clearly a multiplication of continuous symmetry operators corre- • sponds to the addition of their generators in the exponent. The conserved quantum numbers, which are related to F and not to U, are therefore additive. We now have the work plan to find the relation between a contin- • uous symmetry of H and the corresponding conserved observable: 1. Define the symmetry transformation and find the generator F 2. The expectation value of F is a constant of motion Next, we will introduce some concepts of group theory which is • the mathematical framework to systematically describe and clas- sify symmetry operations. 1{9 Exercise 1.1: Show that (consult a quantum mechanics book if necessary) (a) [0:5] Invariance for translations in space leads to the conservation of momentum. (b) [0:5] Invariance for translations in time leads to the conservation of energy. (c) [0:5] Rotational invariance leads to the conservation of angular momentum. 1{10 Group theory It is clear that a combination of two symmetry operations|that • each leaves the system unchanged|is again a symmetry operation. And there is of course the trivial symmetry operation, namely, `do nothing'. Furthermore, we can assume that each symmetry oper- ation can be undone. We say, in fact, that symmetry operations form a group.