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PHY401: Nuclear and

Lecture 24, Wednesday, October 21, 2020 Dr. Anosh Joseph IISER Mohali Discrete Symmetries

Let us look at the following discrete transformations:

Parity P.

Time reversal T.

Charge conjugation C.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Parity (or space inversion) is a transformation that ...

...takes us from a right handed coordinate frame to a left handed one, or vice versa.

Let us denote this by symbol P.

Under this transformation the spacetime four-vector changes as follows

ct  ct   x  −x   P     −→   . (1)  y  −y z −z

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

P is not a subgroup of spatial .

A left handed cannot be obtained from a right handed one through any combination of rotations.

Rotations define a set of continuous transformations.

The inversion of space coordinates does not.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Thus the quantum numbers corresponding to rotations and parity are distinct.

Classically, the components of and vectors change sign under inversion of coordinates,...

... while their magnitudes are preserved

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

P ~r −→ −~r, (2)

P ~p = m~r˙ −→ −m~r˙ = −~p, (3)

√ P p √ r = ~r · ~r −→ −~r · −~r = ~r · ~r = r, (4)

p P p p p = ~p · ~p −→ −~p · −~p = ~p · ~p = p. (5)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

This defines the behavior of normal and vector quantities under P.

There are, however, scalar and vector quantities that do not transform under P as shown in the above equations.

Example: Orbital .

It changes like a vector under a of coordinates, and which we therefore regard as a vector, behaves under P as

P L~ = ~r × ~p −→ (−~r × (−~p) = ~r × ~p = L~ . (6)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

This is, in fact, just opposite of how a normal vector transforms.

Such vectors are consequently called pseudo-vectors or axial vectors.

Similarly, there exists a class of scalars that transform oppositely from normal scalars

P a~ · (~b × ~c) −→ (−a~ ) · (−~b × −~c) = −a~ · (~b × ~c). (7)

Example: volume of a parallelopiped.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Such quantities are known as pseudo-scalars.

Of course, any type of vector can be labeled by one index (namely, by its components).

There are also more complex objects in physics that require more indices, and are known as .

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

The quadrupole moment, the (-momentum) stress , and the relativistic electromagnetic field strength Fµν, are examples of second rank tensors (objects with two indices).

An important property of P: two successive parity transformations leave the coordinate system unchanged.

P P ~r −→ −~r −→ ~r. (8)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

If we think of P as representing the implementing a parity transformation, then from Eq. (8) we conclude that

P2|ψi = +1|ψi. (9)

The eigenvalues of P operator can therefore be only ±1.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

If we have a parity theory, namely,...

... a theory whose Hamiltonian H is invariant under inversion of coordinates, then, P commutes with H

[P, H] = 0. (10)

When P and H commute, the eigenstates of the Hamiltonian are also eigenstates of P, with eigenvalues of either +1 or −1.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

A wave function transforms under P as

P ψ(~r) −→ ψ(−~r). (11)

This implies that the stationary states of any Hamiltonian invariant under a parity transformation have definite parity.

They can be classified as either even or odd functions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Example: One-dimensional harmonic oscillator

It has P invariant Hamiltonian

2 2 p 1 P (−p) 1 H = + mω2x2 −→ + mω2(−x)2 = H. (12) 2m 2 2m 2

We know that the energy eigenstates of the oscillator are Hermite polynomials.

They are either even or odd functions of x, but never a mixture of odd and even functions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Example: A rotationally invariant system in three .

We know that the energy eigenstates in this case are also eigenstates of the angular .

The wave function for the system can be written as

ψnlm (~r) = Rnl(r)Ylm (θ, φ). (13)

Ylm (θ, φ): .

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

P transformation in spherical coordinates takes the form

P r −→ r, (14) P θ −→ π − θ, (15) P φ −→ π + φ. (16)

Under this transformation we have

P l Ylm (θ, φ) −→ Ylm (π − θ, π + φ) = (−1) Ylm (θ, φ). (17)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Consequently, P transforms any wave function that is an eigenstate of orbital angular momentum as

P l ψnlm (~r) −→ (−1) ψnlm (~r). (18)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

In general, a quantum mechanical wave function can have, in addition, an intrinsic parity or phase...

... that is independent of its spatial transformation property of Eq. (19).

Correspondingly, a general that is described by eigenfunctions of orbital angular momentum will transform under P as

P l ψnlm (~r) −→ nψ(−1) ψnlm (~r). (19) nψ: intrinsic parity of the quantum state.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Can think of the intrinsic parity as the phase analog of intrinsic ,...

... which when added to the orbital angular momentum yields the total angular momentum of a system.

As a consequence of Eq. (9), the intrinsic parity satisfies the condition

2 nψ = 1. (20)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Can therefore define a total parity of any such quantum mechanical state as

l ηTOT = nψ(−1) . (21)

A detailed analysis of relativistic quantum theories reveals that...

... bosons have the same intrinsic parities as their , whereas...

... the relative intrinsic parity of and their antiparticles is odd (opposite).

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Newton’s equation of motion for a point particle has the form d2~r m = F~ . (22) dt2

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

If we assume the F~ to be either electromagnetic or gravitational, we can write

C F~ = rˆ. (23) r2 C: a constant.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity

Clearly, since under inversion of coordinates both the left-hand side of Eq. (22) and the right-hand side of Eq. (23) change sign.

Newton’s equation for electromagnetic or gravitational interactions is therefore invariant under space inversion.

It can be shown in a similar fashion that Maxwell’s equations are also invariant under a parity transformation.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

When P is a good symmetry, then the intrinsic parities of different particles can be determined by analyzing different decay or production processes.

However, it is not possible to determine an absolute parity of any system because,...

... starting with some set of assignments, we can invert the parities of all states without observing a physical consequence of that change.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

This is similar, for example, to defining the absolute sign of or other quantum numbers.

A convention is needed to define intrinsic parities of objects that differ in some fundamental way - either through their electric charge, strangeness, or other characteristics.

Accepted convention: choose the intrinsic parities of the , the and the Λ as +1.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

Parities of other particles relative to these assignments can be obtained...

By the analysis of parity-conserving interactions involving such particles.

When parity is conserved, it then restricts the kind of decay processes that can take place.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

Example: Particle A decays in its rest frame into particles B and C A → B + C. (24)

Take J as the spin of the decaying particle.

Then conservation of angular momentum requires that the total angular momentum of the final state also be J.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

In particular, if the two decay products are spinless,...

... then their relative orbital angular momentum (l) must equal the spin of A,

l = J. (25)

Conservation of parity in the decay then implies that

l J ηA = ηBηC(−1) = ηBηc(−1) . (26)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

If the decaying particle has spin zero then we must have ηA = ηBηC. (27)

Hence, the allowed decays correspond to

0+ → 0+ + 0+, (28) 0+ → 0− + 0−, (29) 0− → 0+ + 0−. (30)

J P = 0+ (or 0−): represents the standard convention for labeling a spin and intrinsic parity of a particle.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Conservation

Certain decays are forbidden because they violate parity conservation.

Eg:

+ + − 0 9 0 + 0 , (31) − + + 0 9 0 + 0 , (32) − − − 0 9 0 + 0 . (33)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity of π−

Consider the absorption of very low-energy π− on nuclei

π− + d → n + n. (34)

Denote li and lf as the orbital angular momenta in the initial and final states, respectively.

Then conservation of parity in the reaction would require li lf ηπηd(−1) = ηnηn(−1) . (35)

ηπ, ηd and ηn: intrinsic parities of the three particles.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity of π− Meson

The intrinsic parity of the deuteron is +1. Thus 2 ηn = +1.

It follows that

l −li l +li ηπ = (−1) f = (−1) f . (36)

The capture process is known to proceed from an li = 0 state, and consequently, we get that

l ηπ = (−1) f . (37)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity of π− Meson

Now, with the spin of the deuteron being Jd = 1, this leaves the following possibilities for the state of the two

(1) |ψnn i = |J = 1, s = 1, lf = 0 or 2i, (38) (2) |ψnn i = |J = 1, s = 1, lf = 1i, (39) (3) |ψnn i = |J = 1, s = 0, lf = 1i. (40)

The state with s = 0 corresponds to the antisymmetric singlet spin state (↑↓ − ↓↑).

The state with s = 1 to the symmetric triplet spin state of two neutrons.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity of π− Meson

Because the two neutrons are identical fermions, their overall wave function must be antisymmetric,...

(2) ... which excludes all but |ψnn i from consideration,...

... and specifies that the is a , or has an intrinsic parity of ηπ = −1.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Violation

Until the late 1950s, it was believed that parity was a symmetry of all fundamental interactions.

Namely, physics was believed to be the same whether described in a right handed coordinate system or in a left handed one.

T-D Lee and C-N Yang, undertook a systematic study of all the experimentally known weak decays.

They concluded that there was no evidence supporting conservation of parity in weak processes.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity Violation

They postulated that weak interactions violate parity, and suggested experiments to test their conjecture.

C-S Wu and collaborators showed that Parity was violated in weak interactions involved a study of the β decay of polarized Co-60.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Reversal

Time reversal corresponds to inverting the time axis, or the direction of the flow of time.

In classical mechanics this transformation can be represented as

T t −→ −t, (41) T ~r −→ ~r, (42) T ~p −→ −~p, (43) T L~ −→ −L~ . (44)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Time Reversal

Electromagnetic and gravitational interactions are invariant under time reversal.

By looking at the Schrodinger equation we can show that this equation can be made invariant under T if

T ψ(~r, t) −→ ψ∗(~r, −t). (45)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Time Reversal

Invariance under T implies that the transition amplitudes for the process i → f and the time reversed one f → i have the same magnitude

|Mi→f | → |Mf →i |. (46)

This equation is referred to as the principle of detailed balance.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Time Reversal

The transition rates for the two processes can be quite different.

From Fermi’s Golden Rule, the rates are given by

2π W = |M |2ρ , (47) i→f h i→f f 2π W = |M |2ρ . (48) f →i h f →i i

ρf and ρi : the density of states for the end products in the two reactions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Time Reversal

Time reversal invariance appears to be valid in almost all known fundamental processes.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

Both P and T are discrete spacetime symmetry transformations.

It is natural to ask whether there are any discrete transformations in the internal of a quantum mechanical system.

Charge conjugation is, in fact, this kind of transformation.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

The charge conjugation operation inverts all internal quantum numbers of states, and thereby relates particles to their antiparticles.

We have C Q −→ −Q. (49)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

For a quantum mechanical state we can write

C |ψ(Q,~r, t)i −→ |ψ(−Q,~r, t)i. (50)

Q: represents all the internal quantum numbers such as electric charge, number, number, strangeness etc.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

Two consecutive C transformations will leave a state unchanged.

Thus the eigenvalues of C, or the charge parities of an eigenstate, can be only ±1.

Thus, for example, from

C E~ −→ −E~ , (51) C B~ −→ −B~ , (52) we conclude that the , the quantum of the electromagnetic field, must have a charge parity of −1

ηC(γ) = −1. (53)

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

Can show that Maxwell’s equations do not change under C.

Thus electromagnetic interactions are invariant under C.

But C is violated in weak interactions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

We have

C |νL i −→ |ν¯L i, (54) C |ν¯Ri −→ |νRi. (55)

But there is no evidence for the existence of right handed or left handed antineutrinos.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Charge Conjugation

Consequently, the charge conjugate process of β-decay cannot take place,...

... and charge conjugation therefore cannot be a symmetry of such interactions.

Although both P and C symmetry are violated in β-decay, the combined transformation of CP appears to be a symmetry of such processes.

However, note that CP operation is not a symmetry of all weak interactions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali CPT Theorem

The discrete symmetries P, T and C appear to be violated in some processes.

However, the combined operation of CPT must be a symmetry of essentially any theory that is invariant under Lorentz transformations.

That is, even if the individual transformations do not represent symmetries of any given theory, the product transformation will be a symmetry.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali CPT Theorem

This is known as the CPT theorem.

A consequence of CPT invariance leads to certain very interesting conclusions:

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali CPT Theorem

(1.) Particles satisfy Bose-Einstein statistics when they have integer spin.

They obey Fermi-Dirac statistics when they have half-integer spin.

This has additional implications for relativistic theories: an operator with integer spin must be quantized using commutation relations, and an operator with half-integer spin using anti-commutation relations.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali CPT Theorem

(2.) Particles and their antiparticles have identical and same total lifetimes.

(3.) All the internal quantum numbers of antiparticles are opposite to those of their partner particles.

CPT theorem is consistent with all known observations.

CPT appears to be a true symmetry of all interactions.

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali References

I A. Das and T. Ferbel, Introduction To Nuclear And Particle Physics, World Scientific (2003).

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali End

PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali