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Nuclear and Particle Physics PHY401: Nuclear and Particle Physics 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 symmetry by symbol P. Under this transformation the spacetime four-vector changes as follows 0ct1 0 ct 1 B x C B-xC B C P B C B C −! B C . (1) @ y A @-yA z -z PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity P is not a subgroup of spatial rotations. A left handed coordinate system 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 position and momentum 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 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 scalar 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 angular momentum. It changes like a vector under a rotation 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 tensors. PHY401: Nuclear and Particle Physics Dr. Anosh Joseph, IISER Mohali Parity The quadrupole moment, the (energy-momentum) stress tensor, 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 operator implementing a parity transformation, then from Eq. (8) we conclude that P2j i = +1j i. (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 invariant 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 (~r) −! (-~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 = + m!2x2 −! + m!2(-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 dimensions. We know that the energy eigenstates in this case are also eigenstates of the angular momentum operator. The wave function for the system can be written as nlm (~r) = Rnl(r)Ylm (θ, φ). (13) Ylm (θ, φ): spherical harmonics. 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 nlm (~r) −! (-1) nlm (~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 quantum state that is described by eigenfunctions of orbital angular momentum will transform under P as P l nlm (~r) −! n (-1) nlm (~r). (19) n : 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 spin,... ... 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 n = 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 = n (-1) . (21) A detailed analysis of relativistic quantum theories reveals that... ... bosons have the same intrinsic parities as their antiparticles, whereas... ... the relative intrinsic parity of fermions 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 force 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 electric charge 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 proton, the neutron and the Λ hyperon 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- Meson Consider the absorption of very low-energy π- mesons on deuterium nuclei π- + d ! n + n.
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