P,C, and T SYMMETRIES Project for Advanced Selected Problems in Physics Supervisor: Prof. Dr. Namık Kemal Pak Student: İ. Ufuk

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P,C, and T SYMMETRIES

Project for Advanced Selected Problems in Physics

Supervisor: Prof. Dr. Namık Kemal Pak

Student: İ. UfukTaşdan

Contents

1. Introduction

2. Classical Treatments of Discrete Symmetries

2.1 Parity

2.2 Charge Conjugation

2.3 Time Reversal

3. Relativistic Treatments of Discrete Symmetries

3.1 Parity

3.2 Charge Conjugation

3.3 Time Reversal

3.4 PCT

4. PaisUhlenbeck Oscillator and PCT

5. Conclusion

1. Classical Treatments of Discrete Symmetries

It is proven by Noether’s first theorem that every continuous symmetry in nature

corresponds to some conservation law[3]. For example, since little push on position

vector will not change the motion of physical system, there exist some

constant which we call momentum is a constant of motion. However, it is not

investigated that what would happen, for example, inversing the position vector,

although reversing position does not change motion of physical system. Does it give the

new conservation laws, or is it just a generalization of Noether’s first theorem?

2.1 Parity

The first discrete symmetry that we will cover is space inversion or parity. The

parity operation, as applied to transformation on the coordinate system, changes a

right handed(RH) system into a left handed(LH) system. In other words, take the

mirror image (and turn by 180 in its axis) of a physical motion, say with recorder. The

unbiased viewer(that is new born baby, or a man without common sense, but

knowing the laws of nature) of this recorded film will not understand whether it is a

mirror image or not.

Since we are dealing with quantum mechanics, we work with state kets. How

does an eigenket of the position operator transform under parity ? We claim that

⟩ ⟩

Where is a phase factor and is real. Let us take (since it is insensitive in expectation value of any operator), and subsituting this in (2.1) we have

⟩ ⟩ ; thus , that is we come back to same state by appliying parity operator twice, and actually we want this due to our common sense, that is if we turn one rigid object to its backward, it eventually come to same position. We easily see that from (2.1) is not only unitary but also hermitian,

Its eigenvalue can be only +1 or -1.

Consider a space inverted state, assumed to be obtained by applying a unitary operator known as the parity operator, as follows:

⟩ ⟩

We require the expectation value of x taken with respect to the space inverted state to be opposite in sign.

⟨ | | ⟩ ⟨ ⟩

A very reasonable requirement. This is accomplished if

Where we have used the fact that is unitary. In other words, x and must anticommute. Therefore, every derivative of position with respect to time, i.e. velocity, acceleration etc., will anticommute with .

We should note that,

⟩ ⟩ ⟩

It states that ⟩ is an eigenket of x with eigenvalue –x’, so it must be the same as position eigenket ⟩ up to a phase factor.

Let us consider the momentum operator, since it is mdx/dt in classical manner, it is natural to expect to be odd under parity. Let consider momentum operator as the generator of translation. Since translation followed by parity is equivalent to parity followed translation in the opposite direction, then,

Where is unitary translation operator and it differ from identity operator with

where G is the hermitian generation operator, here we use p as G .

Therefore,

( )

From which follows

{ }

Finally we will consider wave functions under parity transformations. Firstly be a wavefunction of a spinless particle whose state is ⟩;

⟨ ⟩

The wave function of space inverted state, represented by the state ket ⟩ is

⟨ ⟩ ⟨ ⟩

To continue let ⟩is an eigenket of parity. Also, since eigenvalue of parity should be

therefore,

⟩ ⟩

The corresponding wave function will be,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

Since parity inverted state will always take the value +1, whereas unperturbed state eigenvalue depend the wavefunction, therefore ⟩ state is even or odd under parity

However, not all wavefunctions of physical interest with parity, for example in a plane wave which have momentum operator in it, is not expected to be parity eigenket since momentum operator anticommutes with parity operator.

2.2 Charge Conjugation

Classical electrodynamics is unvarying under a change in the sign of all

electric charges ; the potentials and fields reverse their signs, but there is a

compensating charge factor in the Lorentz law, so the forces still come out the

same, for instance

However the lorentz force,

[ ] [ ] [ ]

As a conventional example, consider a charge moving uniformly in a magnetic

field than, suppose we change sign of every particle’s charge suddenly, will it be

go opposite direction? No, since we change the both magnetic field creating

particle’s and spectator’s charge there will be no difference in orbit, or deviation.

Thus, we call ‘changing the sign of the charge’ as charge conjugation, C, and it

converts each particle into its antiparticle:

⟩ ̅⟩

When we state the ‘charge conjugation’ we do not mean to only its electric

charge, for C can be applied to a neutral particle, such as the neutron (yielding an

antineutron), and it changes the sign of all the ‘internal’ quantum numbers -

7 charge, baryon number, lepton number, strangeness, charm, beauty, truth – while leaving mass, energy, momentum, and spin untouched.

As with parity, application of C twice brings us back to the original state:

And hence the eigenvalues of C are . If ⟩ is an eigenstate of C, it follows that,

⟩ ⟩ ̅⟩

So ⟩ and ̅⟩ differ at most by a sign, which means that they represent the same physical state. Thus, only those particles that are their own antiparticles can be eigenstate of C. This leaves us the photon, as well as all those mesons that lie at the center of their Eightfold-Way diagrams: and so on. Note that electromagnetic fields are produced by moving charges that change sign under C, also charge conjugation quantum number is multiplicative, a system of n photons has C eigenvalue (-1)n . For instance, the neutral pion undergoes the decay

And thus has even C-parity ,

⟩ ⟩

It follows that neutral pion cannot decay an odd number of photons.

On the other hand, charge conjugation is not a symmetry of the weak interactions: when applied to a neutrino C gives a left handed antineutrino, which does not occur.[5]

2.2 Time Reversal

What we mean by time reversal is not the objects that come from future, but

it is the reversal of motion. In other words, record some physical event and let

some unbiased viewer watch this film backward will not understand whether the

physical event is foreward or backward. Therefore it should be a symmetry of

nature that we should cover.

Since we do not reverse its position, x will remain same under time reversal,

however its derivatives will not, let be time reversal operator,

Therefore, velocity will take minus sign under time reversal, thus, every

element that have velocity part is odd under time reversal. For example, let apply

this time reversal operation on electromagnetism, the new field will be;

This states that Lorentz Force [ ( ) ]is invariant under time

reversal, and Maxwell Equations under time reversal would be ;

is also invariant under time reversal.

Let us more concentrated on time reversal operator , let ⟩ is

an eigenket of , than,

⟩ ⟩

Where ⟩ is time reversed state, or to be more clear it is motion reversed state. For instance if ⟩ is a momentum eigenstate we expect ⟩ to be equal to ⟩ to a phase factor.

Now, let consider this eigenket ⟩ at t=0 , then a slightly later time , the system would be in,

⟩ ( ) ⟩

Where H is the governing hamiltonian evolving time. Before going too further, let apply time reversal operator first at t=0, and then let system evolve under the influence of the Hamiltonian H. We than have at ,

( ) ⟩

Since we expect that the preceding state ket to be same as

⟩

That is first consider a state ket at earlier time t=- , and reverse the other parts of eigenket. Mathematically ,

( ) ⟩ ( ) ⟩

If it is true for any ket, than,

⟩ ⟩

We know argue that cannot be unitary unless the motion of time reversal is to make sense. Otherwise, suppose to be unitary, than cancel the i’s in (2.43) so that we have,

Consider an energy eigenket ⟩ with energy eigenvalue of En. The corresponding time reversed state would be ⟩ , and we would have,

⟩ ⟩ ⟩

Since it has negative energy value, in other words it states that our free time reversed particle has an energy spectrum of - to 0 which is completely unacceptable. What should we do now? Will we choose to keep going with unphysical relations or improve new mathematical device to get rid of this negative value. For now, let choose latter, because imaginary valued velocities are both unphysical and extraordinary for common sense.

Let A be an operator such that,

⟩ ̌⟩ ⟩ ⟩ ̌⟩ ⟩

And if

⟨ ̌| ̌⟩ ⟨ ⟩

There relations hold, A said to be antiunitary, that is being both unitary and complex conjugate operators in it,

Where U is a unitary operator and K is the coplex conjugate operator that forms the the complex conjugate of any coefficient that multiplies a ket and stand on the right of K. Also if A satisfies the relation of,

⟩ ⟩ ⟩ ⟩

İs said to be antilinear operator. However let more concentrated on K (since it should solve our negative energy value problem). Suppose we have a ket multiplied with some constant c,

⟩ ⟩

Let move on to physical discussion, we have antiunitary operator of time reversal, that solve negative energy problem so that,

Which gives positive energy value.

Relativistic Treatments of Discrete Symmetries

We have seen the classical treatments of discrete symmetries to understand how to

deal with symmetries in the classical quantum mechanics and classical mechanics.

However to understand what is anti-particle, we should work with Dirac’s treatment of

hole theory and eventually relativistic equations of quantum mechanics.

We will use Dirac equation of the form;

( )

Where s are 4x4 gamma matrices of the form ,

( ) ( )

Where s are 2x2 Pauli matrices, and gamma matrices provide and elegant

restatement of the commutation relation,

Where 1 is the 4x4 unit matrix and hereafter will not be explicitly indicated. It is clear

from their definition that the are anti-hermitian, with ( ) , and that is

hermitian. It is convenient to introduce the Feynman dagger, or slash, notation:

And it will be used that,

Than equation (3.1), abbreviates to

( )

Or with ,

Addition of the electromagnetic interaction according to the “minimal” substitution of

( )

Where and is the electric and magnetic potential respectively, and set

Is the required Dirac equation of electron in electro-magnetic field.

3.1 Parity

We now improve our outlook to take into account the existence of the improper

Lorentz transformation of space reflection. Since covariance requires a solution,

The transformation matrix is,

[ ]

Therefore,

Which is satisfied by,

The phase factor would be narrowed to the four choices of or if we require

that four reflections return the spinor to itself in analogy with a rotation through 4

radians. Also, P is evidently unitary. Than the space inverted wavefunction would be,

3.2 Charge Conjugation

If we neglect interaction of the electrons with the radiation field, we may than calculate the stationary state solutions and transition amplitudes. However, the problem of keeping the electron from tumbling into a negative energy states exist in principle, as well as in practice.

We must find some treatment of the negative-energy states, due to Dirac equation being survived, and Dirac did this for us, called “hole theory”, which resolves the dilemma posed by negative-energy solutions simply by filling up the negative-energy levels with electrons, according to Pauli exclusion principle. Therefore, the vacuum state will be one with negative energy levels filled and all positive-energy levels empty The stability of hydrogen atom ground state, for instance, is now guaranteed, since no more electrons can be accommodated in the negative-energy sea by the Pauli principle.

Now, concentrate on radiations, suppose that a negative-energy electron absorb radiation and can be excited into a positive-energy state. If it occurs, we observe an electron of charge -|e| and energy +E and in addition a hole in the negative energy sea. The hole registers the absence of an electron of charge -|e| and Energy –E and that would be interpreted by an observer relative to the vacuum as the presence of a particle of charge

+|e| and energy +E; which we call positron. Therefore, a hole in the negative energy sea recording the absence of an energy -|E| and the absence of -|e|, is equivalent to the presence of a positron with +|E| and +|e|. We therefore have a one-to-one correspondence between the negative-energy solutions of the Dirac equation

( )

and the positron eigenfunctions. Or, equivalently positrons appear as positively charged electrons, the positron wavefunction will be a positive-energy solution of the equation,

( )

Thus, the job is to find an operator transforming the two equations into each other.

We impose an operation of C = [take complex conjugate] x . For instance applying C on

Eq.3.15, will transform this equation into Eq.3.16, and vice versa.

( )

Thus,

( ) ( )

With

For instance, Let us examine for a negative energy electron at rest has a wavefunction of

[ ]

Thus, the corresponding positron solution will be, ,

[ ] [ ]

[ ]

That is the absence of a spin-up negative-energy electron at rest is equivalent to the presence of a spin-down positive energy positron at rest. Applying the same transformation to an arbitrary spin-momentum eigenstate, by using

and

And the momentum projection operator,

( )

And the spin projection operator,

( )

The corresponding positron wavefunction would be,

( ) ( )

Thus C has yielded from a negative energy solution described by four-momentum and polarization a positive-energy solution described by the same .

The surprising and new result to which we have been led by the hole theory is that if there exist electrons of mass m and charge e, there necessarily must also exist positrons of the same mas mm but opposite charge –e.

3.3 Time Reversal

To construct the desired time-reversal transformation, we write the Dirac equation in hamiltonain form,

[ ]

To get rid of negative energy problem, we will use anti-unitary operator as in chapter

2.3. Let ʈ be time reversal operator, than it would has a form of ʈ = [take complex conjugate] x multiply by 4x4 constant matrix T;

Which will give,

[ ]

This means T must commute with and and anticommute with and ; thus

Is satisfactory and the phase factor is again arbitrary.

To show that the transformation ʈ corresponds to what we mean classically by time reversal, we apply ʈ to a plane wave solution for a free particle,

( ) ( ) ( ) ( )

( ) ( )

Where and project a free particle solution with reversed direction of space momentum and spin.

3.4 PCT

Since space and time coordinate inversions P and are invariance operations of the theory, we may just as well include them, if we wish, in constructing the positron wave function. Combining these operators,

Thus an electron wave function multiplied by and moving backward in space- time is just a positron wavefunction. For any free particle spin-momentum eigenstate characterized by , we see that ,

( ) ( )

Therefore, we may picture a positron wavefunction of positive energy as a negative energy electron wave function multiplied by and moving backward in space-time.

For an arbitrary solution in the presence of electromagnetic forces we may explicitely verify this interpretation by returning to the negative-energy eigenvalue equation

[ ]

Than by applying PCT under space-time inversion than the (Eq.3.39) will take the form

[ ]

Which has positive energy. In conclusion we have shown that, we can interpret any positive energy particle to its anti-particle with negative energy and moving backward in space-time.[7]

2. PaisUhlenbeck Oscillator and PCT

In PaisUhlenbeck Oscillator, there was also negative-energy problem which called as

Ostrogradski instability. In Ostrogradski instability some second order Hamiltonian which

is definitely positive valued may sometimes be negative in higher order Hamiltonian.

Moreover, this problem is tried to cured by introducing negative mass particles which is

called ghost particles.

However we have seen that by using charge conjugation operator we can interpret

any negative energy electrons as positive energy positrons. Thus, the main idea that we

will use is to construct some unitary operation such that negative valued Hamiltonian will

be become a positive valued Hamiltonian which satisfies the same equations of motion,

thus we do not need to introduce ghost particles.

The general PaisUhlenbeck oscillator has the form[9]

Where is not gamma matrices here but just constant (to keep citation fixed), and

and are positive and negative energy modes in HOOP (higher order one particle)

viewpoint and and are canonical momentums respectively.

The job begins with to find some unitary operator that change Hamiltonian such that

it would be definitely positive. Since in equation 3.41, part makes

Hamiltonian to be negative, it creates Ostrogradski instability. To make easier, let transform

Than the modified Hamiltonian would be,

Because there exist again term, our Hamiltonian is not definitely positive and instability contunie to exist. As in CPT we will show that our Hamiltonian is PT symmetric if we consider it to be physical. Because Hamiltonian is not hermitian since it has imaginary term in it.

Under PT as shown in preceding chapters momentums keeps its signs whereas positions take minus signs. Thus, under PT only the imaginary part changes it signs which is unimportant when physical property of Hamiltonian considered. Therefore, we are able to interpret ghost as a conventional quantum state of positive PT norm.

If we find some C operator such that

[ ] [ ]

Which indicates C is unitary and surely commute with PT and Hamiltonian. The found

C has the form

Also, and is given by

√

For the Pais-Uhlenbeck Hamiltonian the transformations is;

√ √ √

In the PT-symmetric quantum mechanics, performing a similarity transformation on the PT-symmetric Hamiltonian with yields a positive definite Hamiltonian which is also hermitian. Thus, the new hamiltonian would be

The spectrum of this Hamiltonian is manifestly real and positive. However because this Hamiltonian is related with the original Pais-Uhlanbeck Hamiltonian by a

27 similarity transformation, which is isospectral, despite the –iqx term, the positivity of the PaisUhlenbeckHamiltonain is proved.

Furthermore, the eigenstates of the new Hamiltonian would be,

| ̌

Thus, the normalization gives,

⟨ ̌| ̌⟩ ⟨ ⟩

Finally, because the norm of wavefunction is +1 and because [ ̌ ] , the

Hamiltonian ̌ generates the unitary time evolution.[8]

3. Conclusion

In this project discrete symmetries are investigated. In the classical treatments of

parity or space inversion, charge conjugation and time reversal, it is shown that any

symmetry of nature does not change the equations of motion so that trajectories of

particles are the same. In parity operation we inverted all the possible coordinates twice

and expect to be all eigenvalues of eigenstates are the same, and we found it as it to be.

Also, in charge conjugation part we inverted all the internal properties of particles by

leaving mass, energy, momentum and spin to be the same and results with that only

those particles that are their own antiparticles can be eigenstate of charge conjugation.

Moreover, in time reversal part we use anti-unitary operation to get rid of the negative

energy problem and we have some extraordinary result of being any fermion that is

twice time reversed state would be a different state, and this would be ordinary result if

the rotation about 2 being give again a different state is considered.

In the relativistic treatments of PCT it is found that any antiparticle could be

represented as space-time reverted particle. Also, we showed that “hole theory” make

our negative energy solutions to be useful such that, any particle’s negative energy

solution could be interpreted as an antiparticle’s positive energy solution and vice versa.

Finally we have united the higher order theories with discrete symmetries. We have

resolved the one of the higher order oscillator, PaisUhlenbeck Oscillator’s negative

energy problem or Ostrogradski instability by appliying unitary charge conjugation

operation. In detail, firstly we have showed that our higher order Hamiltonian is

PTsymmetric so that we can find some unitary operation that commutes with both PT and Hamiltonian. After that, we applied this operation to verify PCT symmetry of nature and this makes our Hamiltonian’s spectrum to be real and positive, as we expect from hole theory representation. Therefore we do not need to introduce negative mass- particles so called ghost particles to get rid of the negative energy problem because negative energy solutions have importance on our physical discussions.

References

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“Report of the Working Group on CP Violation and Rare Decays”. Retrieved

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