- A.19.1 An example symmetry transformation
- A.19.2 Physical description of a symmetry
- A.19.3 Derivation of the conservation law
- A.19.4 Other symmetries
- A.19.5 A gauge symmetry and conservation of charge
- A.19.6 Reservations about time shift symmetry

A.19 Conservation Laws and Symmetries

This note has a closer look at the relation between conservation laws and symmetries. As an example it derives the law of conservation of angular momentum directly from the rotational symmetry of physics. It then briefly explains how the arguments carry over to other conservation laws like linear momentum and parity. A simple example of a local gauge symmetry is also given. The final subsection has a few remarks about the symmetry of physics with respect to time shifts.

A.19.1 An example symmetry transformation

The mathematician Weyl gave a simple definition of a symmetry. A symmetry exists if you do something and it does not make a difference. A circular cylinder is an axially symmetric object because if you rotate it around its axis over some arbitrary angle, it still looks exactly the same. However, this note is not concerned with symmetries of objects, but of physics. That are symmetries where you do something, like place a system of particles at a different position or angle, and the physics stays the same. The system of particles itself does not necessarily need to be symmetric here.

As an example, this subsection and the next ones will explore one particular symmetry and its conservation law. The symmetry is that the physics is the same if a system of particles is placed under a different angle in otherwise empty space. There are no preferred directions in empty space. The angle that you place a system under does not make a difference. The corresponding conservation law will turn out to be conservation of angular momentum.

First a couple of clarifications. Empty space should really be understood to mean that there are no external effects on the system. A hydrogen atom in a vacuum container on earth is effectively in empty space. Or at least it is as far as its electronic structure is concerned. The energies associated with the gravity of earth and with collisions with the walls of the vacuum container are negligible. Atomic nuclei are normally effectively in empty space because the energies to excite them are so large compared to electronic energies. As a macroscopic example, to study the internal motion of the solar system the rest of the galaxy can presumably safely be ignored. Then the solar system too can be considered to be in empty space.

Further, placing a system under a different angle may be somewhat awkward. Don’t burn your fingers on that hot sun when placing the solar system under a different angle. And there always seems to be a vague suspicion that you will change something nontrivially by placing the system under a different angle.

There is a different, better, way. Note that you will always need a coordinate system to describe the evolution of the system of particles mathematically. Instead of putting the system of particles under an different angle, you can put that coordinate system under a different angle. It has the same effect. In empty space there is no reference direction to say which one got rotated, the particle system or the coordinate system. And rotating the coordinate system leaves the system truly untouched. That is why the view that the coordinate system gets rotated is called the “passive view.” The view that the system itself gets rotated is called the “active view.”

Figure A.7 shows graphically what happens to the
position coordinates of a particle if the coordinate system gets
rotated. The original coordinate system is indicated by primes. The
polar

angle azimuthal

angle

That is the basic mathematical description of the symmetry transformation.

However, it must still be applied to the description of the physics.
And in quantum mechanics, the physics is described by a wave function

where 1, 2, ..., is the numbering of the particles. Particle spin will be ignored for now.

Physically absolutely nothing changes if the coordinate system is
rotated. So the values

Therefore, considered as functions,

Mathematically, changes in functions are most conveniently written in
terms of an appropriate operator, chapter 2.4. The
operator here is called the “generator of rotations around the

In terms of this operator, the relationship between the wave functions in the rotated and original coordinate systems can be written concisely as

Using

So far, this is all mathematics. The above expression applies whether
or not there is symmetry with respect to rotations. It even applies
whether or not

A.19.2 Physical description of a symmetry

The next question is what it means in terms of physics that empty
space has no preferred directions. According to quantum mechanics,
the Schrödinger equation describes the physics. It says that the
time derivative of the wave function can be found as

where

In particular, consider the two coordinate systems of the previous
subsection. The second system differed from the first by a rotation
over an arbitrary angle

A couple of very basic examples can make this more concrete. Consider
the electronic structure of the hydrogen atom as analyzed in chapter
4.3. The electron was not in empty space in that
analysis. It was around a proton, which was assumed to be at rest at
the origin. However, the electric field of the proton has no
preferred direction either. (Proton spin was ignored). Therefore the
current analysis does apply to the electron of the hydrogen
atom. In terms of Cartesian coordinates, the Hamiltonian in the
original

The first term is the kinetic energy operator. It is proportional to the Laplacian operator, inside the square brackets. Standard vector calculus says that this operator is independent of the angular orientation of the coordinate system. So to get the corresponding operator in the rotated

As a second example, consider the analysis of the complete hydrogen
atom as described in addendum {A.5}. The complete
atom was assumed to be in empty space; there were no external effects
on the atom included. The analysis still ignored all relativistic
effects, including the electron and proton spins. However, it did
include the motion of the proton. That meant that the kinetic energy
of the proton had to be added to the Hamiltonian. But that too is a
Laplacian, now in terms of the proton coordinates

The equality of the Hamiltonians in the original and rotated
coordinate systems has a consequence. It leads to a mathematical
requirement for the operator

That follows from examining the wave function of a system as seen in both the original and the rotated coordinate system. There are two ways to find the time derivative of the wave function in the rotated coordinate system. One way is to rotate the original wave function using

This observation can be inverted to define a symmetry of physics in general:

If an operator commutes with the Hamiltonian, then the same Hamiltonian applies in the changed coordinate system. So there is no physical difference in how systems evolve between the two coordinate systems.A symmetry of physics is described by a unitary operator that commutes with the Hamiltonian.

The qualification “unitary” means that the operator should not change the magnitude of the wave function. The wave function should remain normalized. It does for the transformations of interest in this note, like rotations of the coordinate system, shifts of the coordinate system, time shifts, and spatial coordinate inversions. All of these transformations are unitary. Like Hermitian operators, unitary operators have a complete set of orthonormal eigenfunctions. However, the eigenvalues are normally not real numbers.

For those who wonder, time reversal is somewhat of a special case. To
understand the difficulty, consider first the operation “take
the complex conjugate of the wave function.” This operator
preserves the magnitude of the wave function. And it commutes with
the Hamiltonian, assuming a basic real Hamiltonian. But taking
complex conjugate is not a linear operator. For a linear operator

Now the effect of time-reversal on wave functions turns out to be
antilinear and antiunitary too, [49, p. 76]. One simple
way to think about it is that a straightforward time reversal would
change

A.19.3 Derivation of the conservation law

The definition of a symmetry as an operator that commutes with the
Hamiltonian may seem abstract. But it has a less abstract
consequence. It implies that the eigenfunctions of the symmetry
operation can be taken to be also eigenfunctions of the Hamiltonian,
{D.18}. And, as chapter 7.1.4 discussed, the
eigenfunctions of the Hamiltonian are stationary. They change in time
by a mere scalar factor

The fact that the eigenfunctions do not change is responsible for the
conservation law. Consider what a conservation law really means. It
means that there is some number that does not change in time. For
example, conservation of angular momentum in the

And if the system of particles is described by an eigenfunction of the
symmetry operator, then there is indeed a number that does not change:
the eigenvalue of that eigenfunction. The scalar factor

The eigenvalues of a symmetry of physics describe the possible values of a conserved quantity.

Of course, the system of particles might not be described by a single eigenfunction of the symmetry operator. It might be a mixture of eigenfunctions, with different eigenvalues. But that merely means that there is quantum mechanical uncertainty in the conserved quantity. That is just like there may be uncertainty in energy. Even if there is uncertainty, still the mixture of eigenvalues does not change with time. Each eigenfunction is still stationary. Therefore the probability of getting a given value for the conserved quantity does not change with time. In particular, neither the expectation value of the conserved quantity, nor the amount of uncertainty in it changes with time.

The eigenvalues of a symmetry operator may require some cleaning up.
They may not directly give the conserved quantity in the desired form.
Consider for example the eigenvalues of the rotation operator

The one thing that can be said about the eigenvalues is that they are
always of magnitude 1. Otherwise an eigenfunction would change in
magnitude during the rotation. But a function does not change in
magnitude if it is merely viewed under a different angle. And if the
eigenvalues are of magnitude 1, then the Euler formula
(2.5) implies that they can always be written in the form

where

But although

where the constant of proportionality

The constant magnetic quantum number.

Unfortunately, long before quantum mechanics, classical physics had
already figured out that something was preserved. It called that
quantity the angular momentum

So conservation of angular momentum is the same thing as conservation of magnetic quantum number.

But the magnetic quantum number is more fundamental. Its possible
values are pure integers, unlike those of angular momentum. To see
why, note that in terms of

Now if you rotate the coordinate system over an angle

It may be interesting to see how all this works out for the two
examples mentioned in the previous subsection. The first example was the
electron in a hydrogen atom where the proton is assumed to be at rest
at the origin. Chapter 4.3 found the electron energy
eigenfunctions in the form

It is the final exponential that changes by the expected factor

The second example was the complete hydrogen atom in empty space. In
addendum {A.5}, the energy eigenfunctions were found in
the form

The first term is like before, except that it is computed with a

reduced massthat is slightly different from the true electron mass. The argument is now the difference in position between the electron and the proton. It still produces a factor

As a final step, it is desirable to formulate a nicer operator for
angular momentum. The rotation operators

If you define a rotation operator over a very small angle, call it

Now consider what this operator really means for a single particle
with no spin:

By definition, the final term is the partial derivative of

You can go one better still, because the eigenvalues of the operator
just defined are

If you add a factor

This agrees perfectly with what chapter 4.2.2 got from guessing that the relationship between angular and linear momentum is the same in quantum mechanics as in classical mechanics.

The angular momentum operator of a general system can be defined
using the same scale factor:

Consider now what happens if the angular operator

The limit in the right hand side is a total derivative. According to calculus, it can be rewritten in terms of partial derivatives to give

The scaled derivatives in the new right hand side are the orbital angular momenta of the individual particles as defined above, so

It follows that the angular momenta of the individual particles just add, like they do in classical physics.

Of course, even if the complete system has definite angular
momentum, the individual particles may not. A particle numbered

If every particle has definite momentum like that, then these momenta directly add up to the total system momentum. At the other extreme, if both the system and the particles have uncertain angular momentum, then the expectation values of the momenta of the particles still add up to that of the system.

Now that the angular momentum operator has been defined, the generator
of rotations

(A.77) |

Now consider the generator of rotations in terms of the individual
particles. Since

So, while the contributions of the individual particles to total angular momentum add together, their contributions to the generator of rotations multiply together. In particular, if a particle

How about spin? The normal angular momentum discussed so far suggests
its true meaning. If a particle

But there is something curious here. If the axis system is rotated
over an angle

where the first term has

Now the sign of the wave function does not make a difference for the
observed physics. But it is still somewhat unsettling to see that on
the level of the wave function, nature is only the same when the
coordinate system is rotated over

(Some books now raise the question why the orbital angular momentum functions could not do the same thing. Why could the quantum number of orbital angular momentum not be half-integer too? But of course, it is easy to see why not. If the spatial wave function would be multiple valued, then the momentum operators would produce infinite momentum. You would have to postulate arbitrarily that the derivatives of the wave function at a point only involve wave function values of a single branch. Half-integer spin does not have the same problem; for a given orientation of the coordinate system, the opposite wave function is not accessible by merely changing position.)

A.19.4 Other symmetries

The previous subsections derived conservation of angular momentum from the symmetry of physics with respect to rotations. Similar arguments may be used to derive other conservation laws. This subsection briefly outlines how.

Conservation of linear momentum can be derived from the symmetry of
physics with respect to translations. The derivation is completely
analogous to the angular momentum case. The translation operator

where

For a single particle, this becomes the usual linear momentum operator

It may again be interesting to see how that works out for the two
example systems introduced earlier. The first example was the
electron in a hydrogen atom. In that example it is assumed that
the proton is fixed at the origin. The energy eigenfunctions for
the electron then were of the form

with

is enough to see that. An electron around a stationary proton does not have definite linear momentum. In other words, the linear momentum of the electron is not conserved.

However, the physics of the complete hydrogen atom as described in
addendum {A.5} is independent of coordinate shifts. A
suitable choice of energy eigenfunctions in this context is

where

The derivation of linear momentum can be extended to conduction
electrons in crystalline solids. In that case, the physics of the
conduction electrons is unchanged if the coordinate system is
translated over a crystal period

The conserved quantity in this case is just the

One consequence of the indeterminacy in extended zone scheme,

chapter 6.22.4.
As long as the two reduced zone scheme

description,
chapter 6.22.4.

The time shift operator

where classical physics defines

(A.79) |

Usually, nature is not just symmetric under rotating or translating
it, but also under mirroring it. A transformation that creates a
mirror image of a given system is called a parity transformation. The
mathematically cleanest way to do it is to invert the direction of
each of the three Cartesian axes. That is called spatial inversion.
Physically it is equivalent to mirroring the system using some mirror
passing through the origin, and then rotating the system
18

(In a strictly two-dimensional system, spatial inversion does not work,
since the rotation would take the system into the third dimension. In
that case, mirroring can be achieved by replacing just

The analysis of the conservation law corresponding to spatial
inversion proceeds much like the one for angular momentum. One
difference is that applying the spatial inversion operator a second
time turns

(A.80) |

odd” or “minus oneor

negativeif the eigenvalue is

even” or “oneor

positiveif the eigenvalue is 1.

In a system, the

A.19.5 A gauge symmetry and conservation of charge

Modern quantum theories are build upon so-called “gauge symmetries.” This subsection gives a simple introduction to some of the ideas.

Consider classical electrostatics. The force on charged particles is
the product of the charge of the particle times the so-called electric
field voltage

in
electrical applications. Now it too has a symmetry: adding some
arbitrary constant, call it

Note that this symmetry does not involve the gauges used to measure voltages in any way. Instead it is a reference point symmetry; it does not make a difference what voltage you want to declare to be zero. It is conventional to take the earth as the reference voltage, but that is a completely arbitrary choice. So the term “gauge symmetry” is misleading, like many other terms in physics. A symmetry in a unobservable quantity should of course simply have been called an unobservable symmetry.

There is a relationship between this gauge symmetry in

The problem is that in electrostatics an electron has an electrostatic
energy

Conversely, if the gauge symmetry of the potential is fundamental to
nature, creation of lone charges must be impossible. Each negatively
charged electron that is created must be accompanied by a positively
charged particle so that the net charge that is created is zero. In
electron-positron pair creation, the positive charge

You might of course wonder whether an electrostatic energy
contribution

The gauge symmetry takes on a much more profound meaning in quantum
mechanics. One reason is that the Hamiltonian is based on the
potential, not on the electric field itself. To appreciate the full
impact, consider electrodynamics instead of just electrostatics. In
electrodynamics, a charged particle does not just experience an
electric field vector potential

Here

The gauge property now becomes more general. The constant

produce the exact same electric and magnetic fields as

However, the wave function computed using the potentials

It can be seen by crunching it out that if

(A.81) |

To understand what a stunning result that is, recall the physical
interpretation of the wave function. According to Born, the square
magnitude of the wave function

where

As noted earlier, a symmetry means that you can do something and it
does not make a difference. Since

Geometrically, a complex number like the wave function can be shown in
a two-dimensional complex plane in which the real and imaginary parts
of the number form the axes. Multiplying the number by a factor

For a relatively accessible derivation how the gauge invariance
produces quantum electrodynamics, see [24, pp. 358ff].
To make some sense out of it, chapter 1.2.5 gives a brief
inroduction to relativistic index notation, chapter 12.12 to
the Dirac equation and its matrices, addendum {A.1} to
Lagrangians, and {A.21} to photon wave functions. The

A.19.6 Reservations about time shift symmetry

It is not quite obvious that the evolution of a physical system in empty space is the same regardless of the time that it is started. It is certainly not as obvious as the assumption that changes in spatial position do not make a difference. Cosmology does not show any evidence for a fundamental difference between different locations in space. For each spatial location, others just like it seem to exist elsewhere. But different cosmological times do show a major physical distinction. They differ in how much later they are than the time of the creation of the universe as we know it. The universe is expanding. Spatial distances between galaxies are increasing. It is believed with quite a lot of confidence that the universe started out extremely concentrated and hot at a “Big Bang” about 15 billion years ago.

Consider the cosmic background radiation. It has cooled down greatly since the universe became transparent to it. The expansion stretched the wave length of the photons of the radiation. That made them less energetic. You can look upon that as a violation of energy conservation due to the expansion of the universe.

Alternatively, you could explain the discrepancy away by assuming that
the missing energy goes into potential energy of expansion of the
universe. However, whether this potential energy

is
anything better than a different name for “energy that got
lost” is another question. Potential energy is normally energy
that is lost but can be fully recovered. The potential energy of
expansion of the universe cannot be recovered. At least not on a
global scale. You cannot stop the expansion of the universe.

And a lack of exact energy conservation may not be such a bad thing for physical theories. Failure of energy conservation in the early universe could provide a possible way of explaining how the universe got all that energy in the first place.

In any case, for practical purposes nontrivial effects of time shifts seem to be negligible in the current universe. When astronomy looks at far-away clusters of galaxies, it sees them as they were billions of years ago. That is because the light that they emit takes billions of years to reach us. And while these galaxies look different from the current ones nearby, there is no evident difference in their basic laws of physics. Also, gravity is an extremely small effect in most other physics. And normal time variations are negligible compared to the age of the universe. Despite the Big Bang, conservation of energy remains one of the pillars on which physics is build.