P,C, and T SYMMETRIES Project for Advanced Selected Problems in - PDF document

P,C, and T SYMMETRIES Project for Advanced Selected Problems in Physics Supervisor: Prof. Dr. Namık Kemal Pak Student: İ. UfukTaşdan 1

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 2

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 ⟩ ⟩ 3

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 Or 4

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 md x /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 { } 5

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. 6

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. 8

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] 9

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, 10

⟩ ⟩ 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, 11

⟩ ⟩ 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 E n . 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 ̌| ̌⟩ ⟨ ⟩ ⟨ 12