1 Parity. 2 Time reversal. Even. Odd. Symmetry Lecture 9

Transcription

1 Even Odd Symmetry Lecture 9 1 Parity The normal mode of a tring have either even or odd ymmetry. Thi alo occur for tationary tate in Quantum Mechanic. The tranformation i called partiy. We previouly found for the harmonic ocillator that there were 2 ditinct type of wave function olution characterized by the election of the tarting integer in their erie repreentation. Thi election produced a erie in odd or even power of the coordiante o that the wave function wa either odd or even upon reflection about the origin, x = 0. Since the potential energy function depend on the quare of the poition, x 2, the energy eignevalue wa alway poitive and independent of whether the eigenfunction were odd or even under reflection. In 1-D parity i the ymmetry operation, x x. In 3-D the trong interaction i invarient under the ymmetry of parity. r r Parity i a mirror reflection plu a rotation of 180, and tranform a right-handed coordinate ytem into a left-handed one. Our Macrocopic world i clearly handed, but handedne in fundamental interaction i more involved. Vector (tenor of rank 1), a illutrated in the definition above, change ign under Parity. Scalar (tenor of rank 0) do not. One can then contruct, uing tenor algebra, new tenor which reduce the tenor rank and/or change the ymmetry of the tenor. Thu a dual of a ymmetric tenor of rank 2 i a peudovector (cro product of two vector), and a calar product of a peudovector and a vector create a peudocalar. We will contruct bilinear form below which have thee rotational and reflection characteritic. 2 Time reveral Time reveral i the mathematical operation; 1

2 t t; with the exchange of initial and final tate. Macrocopically T i not a good ymmetry. However, For Quantum Mechanic; T HT = H; Hψ = i t ψ; THψ = [Tψ]; H[Tψ] = i [Tψ]. Thu Ψ and [T ψ] are not equivalent, and T require t t and i i. However, one contruct obervable in Quantum Mechanic by bilinear form, ( i.e. by product two operator and wave function) o that microcopic time reveribility hold. 3 Charge conjugation Charge Conjugation change a particle to it anti-particle, but without change to it dynamical variable. The ymmetry i baed on the aumption that for every particle there i an antiparticle which ha Q Q, B B, L L, etc. An eigentate of C mut have: Q = B = L = S 0. Thu a π 0 i an eigentate of C but K 0 i not ince it contain S, a quark of the 2 nd generation. The trong and electromagnetic interaction are invarient under C. Under the weak interaction the operation C i not a good ymmetry. 4 The operation of P and T The operation of reflection and time reveral in claical ytem i hown in Table below. 2

3 µ + e + _ C Violation in ν e νµ e + e _ ν e ν µ µ + µ µ C CP _ ν e ν µ e _ ν νµ e Name P T Time + - Poition - + Energy + + Momentum - - Spin + - Helicity - + Electric Field - + Magnetic Field + - Obviouly ome parameter are invarient but ome change ign under thi combined operation. 5 The operation of P and C The operation of CP i compoed of the imultaneou operation of C and P. Suppoe one wihe to ditinguih a galaxy from an anti-galaxy. It i not ufficient to find C violation but one need CP violation a well. The weak interaction violate C and P but CP i experimentally conerved except for flavor changing decay. In flavor changing weak decay CP i not preerved. The K 0 and K 0 are eigentate of trangene but not of CP. However tate of the weak interaction ( a preently defined) are invarient under CP. Thu the two poible CP eigentate of the K 0 (K 0 ) have different mae and decay width. However it wa found that the decay of CP eigentate doe not preerve CP. 3

4 C and P Tranformation for π + µ + ν µ ν π + µ µ + µ + π + P ν µ C C CP _ ν µ π π _ µ P µ ν µ 6 The operation of P and T The operation of reflection and time reveral in claical ytem i hown in Table below. Name P T Time + - Poition - + Energy + + Momentum - - Spin + - Helicity - + Electric Field - + Magnetic Field + - Obviouly ome parameter are invarient but ome change ign under thi combined operation. 7 Bilinear form of Dirac wave function We recall that a Dirac wave function ha 4-component, and that γ, α and β are 4 4 matricie ued in the dirac equation. A an aide note that the current j = cψ αψ lead to an expectation value of the velocity. We write the following bilinear form that have the variou lited tranformation properite; In the above, ( 0 σ γ = σ 0 ) 4

5 Bilinear Form Tranformation Property ψ ψ ψ γ n ψ ψ γ 5 ψ ψ γ 5 γ n ψ Scalar Vector Peudocalar Peudovector State Energy Helicity Chirality 1 > > < < ( ) I 0 γ 0 = 0 I ( ) 0 I γ 5 = i I 0 where σ are the Pauli pin matricie, I i the 2 2 idenity matrix, and γ 5 = γ 1 γ 2 γ 3 γ 5. Note that the γ i are the component of a relativitic 4-vector, and ψ i the adjoint of the Dirac wave function ψ. The helicity of the wave function i defined a the direction of the particle pin vector relative to the momentum vector. It meaure the handedne of a particle and i a peudocalar invarient under T. Σ = σ p/ p The chirality operator, γ 5, operate on the helicity tate to produce the chirality of the tate. We then find the helicity and chirality of the eigenvector for the variou Dirac tate For E > 0 tate a patial reflection invert the momentum vector and change the ign of the helicity. The Chirality of a tate i optained by the projection operator; 5

6 P ± = 1/2(1 ± iγ 5 ) The projection operator ha the propertie; P + + P = 0 P 2 ± = 1 P + P = P P + = 0 8 The operation of P, C, and T Conervation of the imultaneou application of C, P, and T i expected under very general condition. In all Lorentz invarient quantum field theorie, CPT i a good ymmetry. Thi mean that if CP i violated then T mut be violated a well. Direct earche for T violation are difficult a null experiment are not eaily deigned. 9 Zero ma equation In the cae of zero ma the Dirac equation ha the form; [i t + iγ 0 γ ]ψ = 0 Which look like ; ( ) 0 σ [E ]ψ = 0 σ 0 If we divide the 4-component Dirac wave function into 2 two component wave function decribed by upper and lower component ψ u and ψ l repectively, the dirac equation when the ma = 0 form 2 equation; Eψ u σ p ψ l = 0 Eψ l σ p ψ u = 0 Then a pecific chirality tate i not a tate of pecific parity, and can be decribed by a 2-component wave function; 6

7 Symmetry Operation + P T + C + 7

8 ( ) 0 ψ = P ψ = φ ( ) φ ψ + = P + ψ = 0 and ; γ 5 ψ = ψ γ 5 ψ + = ψ + Thu chirality i a good ymmetry for male particle. It repreent the direction of the pin relative to the momentum vector, and divide mael Dirac tate into left and right handed doublet. 10 Lagrangian The lagrangian which i a Lorentz calar, but be compoed of bilinear form in a way to make a calar. Thu for example a vector form mut be contracted (dot product) with a vector. Two peudo-calar can be multipled together, etc. 8