PHY304 | Particle Physics | Dr C N Booth |
Without invariance principles, there would be no laws of physics!
We rely on the results of experiments remaining the same from day to day
and place to place. An invariance principle reflects a basic
symmetry, and is always intimately related to a conservation law (and
to a quantity that cannot be determined absolutely).
The proofs presented in the lectures rely on some
key results from quantum mechanics.
Some classical invariance principles are related to the nature of space-time. Invariance of the Hamiltonian (the operator or expression for total energy) under a translation for an isolated, multiparticle system leads directly to the conservation of the total momentum of the system. This can be demonstrated classically, but we will take a quantum mechanical approach, defining an operator D̂ which produces a translation of the wavefunction through δx:
Then it can be shown that | D̂ ≡ exp(iP̂δx/ħ) | where P̂ is the momentum operator. |
P̂ is said to act as a
"generator of translations". Now since the energy of an isolated
system cannot be affected by a translation of the whole system, D̂
must commute with the Hamiltonian operator Ĥ,
i.e.
We therefore have three equivalent statements:
Another conserved quantity is electric charge, corresponding to an invariance
of physical systems under a translation in the electrostatic potential.
In classical electrostatics, absolute potential is arbitrary - the physics only depends on potential differences.
Assuming this fact remains true, we can consider what would be the consequences of the possibility of
creating and destroying electric charge.
By hypothesis, the energy required to create a charge Q at a potential Φ_{1}
would be W, independent of Φ.
But we could move the charge to another point at Φ_{2}, liberating an energy
Quantum mechanically, we may define a charge operator Q̂ which, when it operates on a wavefunction ψ_{q} describing a system of total charge q, returns an eigenvalue of q.
The above continuous transformations led to additive conservation laws - the sum of all charges or momenta is conserved. There are also discrete or discontinuous transformations, which lead to multiplicative conservation laws. An important group of these are parity P, charge conjugation C and time reversal T.
The parity operator inverts spatial coordinates. It therefore
transforms x into −x, p into −p
etc.
In other words, polar vectors change sign; axial
vectors, such as angular momentum J, do not.
Now | P ψ(x) = ψ(−x). | |
P^{2} ψ(x) = P ψ(−x) = ψ(x). | ||
The parity operator thus has eigenvalues of ±1. | ||
If | P ψ(x) = +ψ(x) | the wavefunction is said to have even parity, |
while if | P ψ(x) = −ψ(x) | it has odd parity. |
The spherical harmonics Y_{l}^{m}(θ, φ) (met in atomic physics and elsewhere) are examples of eigenfunctions of the parity operator. (If they are not familiar, look them up!) By considering a reflection in the origin, it should be clear that in spherical polar coordinates, the parity operator causes
i.e. | P Y_{l}^{m} = (−1)^{l} Y_{l}^{m}. |
Invariance with respect to P leads to multiplicative conservation laws.
E.g. Consider | a + b → c + d |
π_{a} π_{b} (−1)^{l} = π_{c} π_{d} (−1)^{l'} | where l' is the final relative angular momentum. |
Another discrete transformation is charge conjugation, C, which changes a particle into its antiparticle. This reverses the charge, magnetic moment, baryon number and lepton number of the particle.
Time reversal, T reverses the time coordinate.
However, as will be shown in the lectures, T does not satisfy the simple eigenvalue equation
(In fact, defining T like this not only does not have the desired effect of causing momentum to be reversed while
leaving energy unchanged, it results in a wavefunction which does not obey Schrodinger's equation.)
Instead T must be defined by
The strong and electromagnetic interactions are invariant under C, P and T transformations. This is not true of the weak interaction, as can be seen by considering neutrinos (which are only involved in weak interactions). Neutrinos are always left-handed, i.e. their spin is antiparallel to their direction of motion. The P operator reverses momentum but not spin, so when applied to a neutrino would produce a right-handed neutrino, which is not observed, Similarly C applied to a neutrino produces an unobserved left-handed antineutrino. Weak interactions therefore violate C and P. The combination CP, however, applied to a left-handed neutrino produces a right-handed antineutrino, which is observed. Therefore (to a good approximation) weak interactions are invariant under the combined transformation CP. The weak interaction, and all other interactions, are exactly invariant under the combination CPT.
Summary
Invariance | Conserved Quantity |
Gravitation, weak, electromagnetic and strong interactions are independent of: | |
translation in space | linear momentum |
rotations in space | angular momentum |
translations in time | energy |
EM gauge transformation | electric charge |
CPT | (product of parities below) |
Gravitation, electromagnetic and strong interactions are independent of: | |
spatial inversion P | spatial parity |
charge conjugation C | "charge parity" |
time reversal T | "time parity" |
For examples of calculations involving the parity of a multi-particle state, see homework 4.
Supplementary Material
For further, non-technical reading, you might like to consult
The Ideas of Particle Physics,
chapter 6.
More information on CP violation is available from the following source.