PHY304 Particle Physics Dr C N Booth

Invariance Principles and Conservation Laws

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:

D̂ψ(x)  = ψ'(x) = ψ(x + 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. [D̂, Ĥ] = 0; it must therefore also be true that [P̂, Ĥ] = 0, and so P̂ has eigenvalues which are constants of the motion.

We therefore have three equivalent statements:

  1. Momentum is conserved in an isolated system.
  2. The Hamiltonian is invariant under spatial translations.  (Equivalently, it is impossible to determine absolute positions.)
  3. The momentum operator commutes with the Hamiltonian.

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 Φ1 would be W, independent of Φ. But we could move the charge to another point at Φ2, liberating an energy 1 − Φ2)Q, before destroying it to release W: a net energy gain of 1 − Φ2)Q. The ability to create or destroy charge thus violates conservation of energy. Inverting the argument, conservation of energy together with invariance with respect to a change in electric potential automatically requires charge to be conserved. Again, an invariance principle implies a conservation law.

Quantum mechanically, we may define a charge operator Q̂ which, when it operates on a wavefunction ψq describing a system of total charge q, returns an eigenvalue of q.

Q̂ψq = qψq
If q is conserved, Q̂ and Ĥ must commute, and (as will be shown in the lecture) this is assured by invariance under a global phase (or gauge) transformation
ψ'q = exp(iεQ̂)ψq
where ε is an arbitrary real parameter. This is very closely analogous to the to the relationship between conservation of momentum and invariance under displacement, as will be demonstrated in the lectures. (Invariance under a local gauge transformation, where ε can depend on space and time, has much greater consequences, but that is beyond the scope of this lecture course.)

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).
P2 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.
(Note that wavefunctions do not have to be eigenfunctions of parity.)

The spherical harmonics Ylm(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

rr (unchanged)
θ → π − θ
φ → π + φ,
and by inspection of the form of the spherical harmonics it can be seen that Ylm changes sign if l is odd and remains the same if it is even,
i.e. P Ylm = (−1)l Ylm.

Invariance with respect to P leads to multiplicative conservation laws.
E.g. Consider a + b → c + d
The initial state wavefunction can be written as ψi = ψa ψbψl, where ψa and ψb are "internal" wavefunctions for particles a and b, and ψl is the wavefunction describing their relative motion. The P operator affects each factor, so

i = Pψabl
If the intrinsic parities of the particles are given by a = πa ψa, etc., where πa is just a number ±1, then
i = πa πb (−1)l ψi     or    πi = πa πb (−1)l
i.e. the parity of a multiparticle system is given by the product of the intrinsic parities of the individual particles and the parity of the wavefunction describing their relative motion.
A similar expression can be written for the final state. Thus, if the interaction responsible for the above process is invariant under parity (as the electromagnetic interaction is) then
πa πb (−1)l = πc πd (−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 T ψ(t) = ψ(−t) = aψ(t).
(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

T ψ(t) = ψ*(−t).

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.


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.

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