1 Parity 2 Time Reversal

Even Odd

Symmetry Lecture 9

The normal modes of a string have either even or odd symmetry. This also occurs for stationary states in Quantum Mechanics. The transformation is called partiy. We previously found for the harmonic oscillator that there were 2 distinct types of wave function solutions characterized by the selection of the starting integer in their series representation. This selection produced a series in odd or even powers of the coordiante so that the wave function was either odd or even upon reﬂections about the origin, x = 0. Since the potential energy function depends on the square of the position, x2, the energy eignevalue was always positive and independent of whether the eigenfunctions were odd or even under reﬂection. In 1-D parity is the symmetry operation, x → −x. In 3-D the strong interaction is invarient under the symmetry of parity.

~r → −~r

Parity is a mirror reflection plus a rotation of 180◦, and transforms a right-handed coordinate system into a left-handed one. Our Macroscopic world is clearly “handed”, but “handedness” in fundamental interactions is more involved. Vectors (tensors of rank 1), as illustrated in the definition above, change sign under Parity. Scalars (tensors of rank 0) do not. One can then construct, using tensor algebra, new tensors which reduce the tensor rank and/or change the symmetry of the tensor. Thus a dual of a symmetric tensor of rank 2 is a pseudovector (cross product of two vectors), and a scalar product of a pseudovector and a vector creates a pseudoscalar. We will construct bilinear forms below which have these rotational and reflection char- acteristics.

2 Time reversal

Time reversal is the mathematical operation;

1 t → −t;

with the exchange of initial and ﬁnal states. Macroscopically T is not a good symmetry. However, For Quantum Mechanics;

T †HT = H; ∂ Hψ = i~ ψ; ∂t T Hψ = [T ψ]; H[T ψ] = −i~ [T ψ].

Thus Ψ and [T †ψ] are not equivalent, and T † requires t → −t and i → −i. However, one constructs observables in Quantum Mechanics by bilinear forms, ( i.e. by products two operators and wave functions) so that microscopic time reversibility holds.

3 Charge conjugation

Charge Conjugation changes a particle to its anti-particle, but without change to its dynam- ical variables. The symmetry is based on the asumption that for every particle there is an antiparticle which has Q → −Q, B → −B, L → −L, etc.

An eigenstate of C must have:

Q = B = L = S ··· 0.

Thus a π0 is an eigenstate of C but K0 is not since it contains S, a quark of the 2nd generation. The strong and electromagnetic interactions are invarient under C. Under the weak interaction the operation C is not a good symmetry.

4 The operations of P and T

The operations of reﬂection and time reversal in classical systems is shown in Table below.

2 _ µ+ + ν C Violation in e e νµ _ ν νµ e+ e− e s s + s s µ µ− µ− s Cs CP s s s s _ s _ s ν ν ν νµ − e µ e e

Name P T Time + - Position - + Energy + + Momentum - - Spin + - Helicity - + Electric Field - + Magnetic Field + - Obviously some parameters are invarient but some change sign under this combined operation.

5 The operations of P and C

The operation of CP is composed of the simultaneous operations of C and P . Suppose one wishes to distinguish a galaxy from an anti-galaxy. It is not suﬃcient to ﬁnd C violation but one needs CP violation as well.

The weak interaction violates C and P but CP is experimentally conserved except for ﬂavor changing decays. In ﬂavor changing weak decay CP is not preserved.

The K0 and K0 are eigenstates of strangeness but not of CP. However states of the weak interaction ( as presently deﬁned) are invarient under CP. Thus the two possible CP eigenstates of the K0 (K0) have diﬀerent masses and decay widths. However it was found that the decay of CP eigenstates does not preserve CP.

3 + + C and P Transformations for π µ νµ

+ sπ µ+ µ + π+ s ν νµ P µ

CC CP _ _ − − − − ν νµ π µ P µ π µ s s

6 The operations of P and T

The operations of reﬂection and time reversal in classical systems is shown in Table below.

7 Bilinear forms of Dirac wave functions

We recall that a Dirac wave function has 4-components, and that ~γ, ~α and β are 4 × 4 matricies used in the dirac equation. As an aside note that the current ~j = cψ~αψ leads to an expectation value of the velocity. We write the following bilinear forms that have the various listed transformation properites;

In the above,

0 −~σ ~γ = ~σ 0

4 Bilinear Form Transformation Property

ψ ψ Scalar ψγn ψ Vector ψγ5 ψ Pseudoscalar ψγ5γn ψ Pseudovector

State Energy Helicity Chirality

1 > 0 +1 +1 2 > 0 -1 -1 3 < 0 +1 -1 4 < 0 -1 +1

−I 0 γ0 = 0 I 0 I γ5 = −i I 0

where ~σ are the Pauli spin matricies, I is the 2 × 2 idenity matrix, and γ5 = γ1γ2γ3γ5. Note that the γi are the components of a relativistic 4-vector, and ψ is the adjoint of the Dirac wave function ψ.

The helicity of the wave function is deﬁned as the direction of the particle spin vector relative to the momentum vector. It measures the handedness of a particle and is a pseudoscalar invarient under T .

Σ = ~σ · ~p/|~p|

The chirality operator, γ5, operates on the helicity states to produce the chirality of the state. We then ﬁnd the helicity and chirality of the eigenvectors for the various Dirac states

For E > 0 states a spatial reﬂection inverts the momentum vector and changes the sign of the helicity. The Chirality of a state is optained by the projection operator;

5 P± = 1/2(1 ± iγ5)

The projection operator has the properties;

P+ + P− = 0 2 P± = 1 P+P− = P−P+ = 0

8 The operations of P, C, and T

Conservation of the simultaneous application of C, P, and T is expected under very general conditions. In all Lorentz invarient quantum ﬁeld theories, CPT is a good symmetry. This means that if CP is violated then T must be violated as well. Direct searches for T violation are diﬃcult as null experiments are not easily designed.

9 Zero mass equation

In the case of zero mass the Dirac equation has the form;

∂ [i + iγ0~γ · ∇~ ]ψ = 0 ∂t Which looks like ;

0 ~σ [E − ]ψ = 0 σ 0 If we divide the 4-component Dirac wave function into 2 two component wave functions described by upper and lower components ψu and ψl respectively, the dirac equation when the mass = 0 forms 2 equations;

Eψu − ~σ · ~pψl = 0

Eψl − ~σ · ~pψu = 0

Then a speciﬁc chirality state is not a state of speciﬁc parity, and can be described by a 2-component wave function;

6 Symmetry Operations

+ −

− P +

+ +

− T −

+ −

− C +

7 0 ψ = P ψ = − − φ φ ψ+ = P+ ψ = 0 and ;

γ5 ψ− = ψ−

γ5 ψ+ = −ψ+

Thus chirality is a good symmetry for massless particles. It represents the direction of the spin relative to the momentum vector, and divides masselss Dirac states into left and right handed doublets.

10 Lagrangian

The lagrangian which is a Lorentz scalar, but be composed of bilinear forms in a way to make a scalar. Thus for example a vector form must be contracted (dot product) with a vector. Two pseudo-scalars can be multipled together, etc.