7.3 Conservation Laws and Symmetries

Physical laws like conservation of linear and angular momentum are important. For example, angular momentum was key to the solution of the hydrogen atom in chapter 4.3. More generally, conservation laws are often the central element in the explanation for how simple systems work. And conservation laws are normally the most trusted and valuable source of information about complex, poorly understood, systems like atomic nuclei.

It turns out that conservation laws are related to fundamental symmetries of physics. A symmetry means that you can do something that does not make a difference. For example, if you place a system of particles in empty space, far from anything that might affect it, it does not make a difference where exactly you put it. There are no preferred locations in empty space; all locations are equivalent. That symmetry leads to the law of conservation of linear momentum. A system of particles in otherwise empty space conserves its total amount of linear momentum. Similarly, if you place a system of particles in empty space, it does not make a difference under what angle you put it. There are no preferred directions in empty space. That leads to conservation of angular momentum. See addendum {A.19} for the details.

Why is the relationship between conservation laws and symmetries important? One reason is that it allows for other conservation laws to be formulated. For example, for conduction electrons in solids all locations in the solid are not equivalent. For one, some locations are closer to nuclei than others. Therefore linear momentum of the electrons is not conserved. (The total linear momentum of the complete solid is conserved in the absence of external forces. In other words, if the solid is in otherwise empty space, it conserves its total linear momentum. But that does not really help for describing the motion of the conduction electrons.) However, if the solid is crystalline, its atomic structure is periodic. Periodicity is a symmetry too. If you shift a system of conduction electrons in the interior of the crystal over a whole number of periods, it makes no difference. That leads to a conserved quantity called crystal momentum, {A.19}. It is important for optical applications of semiconductors.

Even in empty space there are additional symmetries that lead to important conservation laws. The most important example of all is that it does not make a difference at what time you start an experiment with a system of particles in empty space. The results will be the same. That symmetry with respect to time shift gives rise to the law of conservation of energy, maybe the most important conservation law in physics.

In a sense, time-shift symmetry is already built-in into the Schrö­din­ger equation. The equation does not depend on what time you take to be zero. Any solution of the equation can be shifted in time, assuming a Hamiltonian that does not depend explicitly on time. So it is not really surprising that energy conservation came rolling out of the Schrö­din­ger equation so easily in section 7.1.3. The time shift symmetry is also evident in the fact that states of definite energy are stationary states, section 7.1.4. They change only trivially in time shifts. Despite all that, the symmetry of nature with respect to time shifts is a bit less self-evident than that with respect to spatial shifts, {A.19}.

As a second example of an additional symmetry in empty space, physics works, normally, the same when seen in the mirror. That leads to a very useful conserved quantity called parity. Parity is somewhat different from momentum. While a component of linear or angular momentum can have any value, parity can only be 1, called even,” or $\vphantom0\raisebox{1.5pt}{$-$}$1, called “odd. Also, while the contributions of the parts of a system to the total momentum components add together, their contributions to parity multiply together, {A.19}. That is why the $\pm1$ parities of the parts of a system can combine together into a corresponding system parity that is still either 1 or $\vphantom0\raisebox{1.5pt}{$-$}$1.

(Of course, there can be uncertainty in parity just like there can be uncertainty in other quantities. But the measurable values are either 1 or $\vphantom0\raisebox{1.5pt}{$-$}$1.)

Despite having only two possible values, parity is still very important. In the emission and absorption of electromagnetic radiation by atoms and molecules, parity conservation provides a very strong restriction on which electronic transitions are possible. And in nuclear physics, it greatly restricts what nuclear decays and nuclear reactions are possible.

Another reason why the relation between conservation laws and symmetries is important is for the information that it produces about physical properties. For example, consider a nucleus that has zero net angular momentum. Because of the relationship between angular momentum and angular symmetry, such a nucleus looks the same from all directions. It is spherically symmetric. Therefore such a nucleus does not respond in magnetic resonance imaging. That can be said without knowing all the complicated details of the motion of the protons and neutrons inside the nucleus. So-called spin $\leavevmode\kern.03em
\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ nuclei have the smallest possible nonzero net angular momentum allowed by quantum mechanics, with components that can be $\pm\frac12\hbar$ in a given direction. These nuclei do respond in magnetic resonance imaging. But because they depend in a relatively simple way on the direction from which they are viewed, their response is relatively simple.

Similar observations apply for complete atoms. The hydrogen atom is spherically symmetric in its ground state, figure 4.9. Although that result was derived ignoring the motion and spin of the proton and the spin of the electron, the hydrogen atom remains spherically symmetric even if these effects are included. Similarly, the normal helium atom, with two electrons, two protons, and two neutrons, is spherically symmetric in its ground state. That is very useful information if you want, say, an ideal gas that is easy to analyze. For heavier noble gases, the spherical symmetry is related to the “Ramsauer effect” that makes the atoms almost completely transparent to electrons of a certain wave length.

As you may guess from the fact that energy eigenstates are stationary, conserved quantities normally have definite values in energy eigenstates, {A.19.3}. (An exception may occur when the energy does not depend on the value of the conserved quantity.) For example, nuclei, lone atoms, and lone molecules normally have definite net angular momentum and parity in their ground state. Excited states too will have definite angular momentum and parity, although the values may be different from the ground state.

It is also possible to derive physical properties of particles from their symmetry properties. As an example, addendum {A.20} derives the spin and parity of an important class of particles, including photons, that way.

Finally, the relation between conservation laws and symmetries gives more confidence in the conservation laws. For example, as mentioned nuclei are still poorly understood. It might therefore seem reasonable enough to conjecture that maybe the nuclear forces do not conserve angular momentum. And indeed, the force between the proton and neutron in a deuteron nucleus does not conserve orbital angular momentum. But it is quite another matter to suppose that the forces do not conserve the net angular momentum of the nucleus, including the spins of the proton and neutron. That would imply that empty space has some inherent preferred direction. That is much harder to swallow. Such a preferred direction has never been observed, and there is no known mechanism or cause that would give rise to it. So physicists are in fact quite confident that nuclei do conserve angular momentum just like everything else does. The deuteron conserves its net angular momentum if you include the proton and neutron spins in the total.

It goes both ways. If there is unambiguous evidence that a supposedly conserved quantity is not truly conserved, then nature does not have the corresponding symmetry. That says something important about nature. This happened for the mirror symmetry of nature. If you look at a person in a mirror, the heart is on the other side of the chest. On a smaller scale, the molecules that make up the person change in their mirror images. But physically the person in the mirror would function just fine. (As long as you do not try to mix mirror images of biological molecules with nonmirror images, that is.) In principle, evolution could have created the mirror image of the biological systems that exist today. Maybe it did on a different planet. The electromagnetic forces that govern the mechanics of biological systems obey the exact same laws when nature is seen in the mirror. So does the force of gravity that keeps the systems on earth. And so does the so-called strong force that keeps the atomic nuclei together.

Therefore it was long believed that nature behaved in exactly the same way when seen in the mirror. That then leads to the conserved quantity called parity. But eventually, in 1956, Lee and Yang realized that the decay of a certain nuclear particle by means of the so-called weak force does not conserve parity. As a result, it had to be accepted also that nature does not always behave in exactly the same way when seen in the mirror. That was confirmed experimentally by Wu and her coworkers in 1957. (In fact, while other experimentalists like Lederman laughed at the ideas of Lee and Yang, Wu spend eight months of hard work on the risky proposition of confirming them. If she had been a man, she would have been given the Nobel Prize along with Lee and Yang. However, Nobel Prize committees have usually recognized that giving Nobel Prizes to women might interfere with their domestic duties.)

Fortunately, the weak force is not important for most applications, not even for many involving nuclei. Therefore conservation of parity usually remains valid to an excellent approximation.

Mirroring corresponds mathematically to an inversion of a spatial coordinate. But it is mathematically much cleaner to invert the direction of all three coordinates, replacing every position vector ${\skew0\vec r}$ by $\vphantom0\raisebox{1.5pt}{$-$}$${\skew0\vec r}$. That is called “spatial inversion.” Spatial inversion is cleaner since no choice of mirror is required. That is why many physicists reserve the term “parity transformation” exclusively to spatial inversion. (Mathematicians do not, since inversion does not work in strictly two-di­men­sion­al systems, {A.19}.) A normal mirroring is equivalent to spatial inversion followed by a rotation of 180$\POW9,{\circ}$ around the axis normal to the chosen mirror.

Inversion of the time coordinate is called “time reversal.” That can be thought of as making a movie of a physical process and playing the movie back in reverse. Now if you make a movie of a macroscopic process and play it backwards, the physics will definitely not be right. However, it used to be generally believed that if you made a movie of the microscopic physics and played it backwards, it would look fine. The difference is really not well understood. But presumably it is related to that evil demon of quantum mechanics, the collapse of the wave function, and its equally evil macroscopic alter ego, called the second law of thermodynamics. In any case, as you might guess it is somewhat academic. If physics is not completely symmetric under reversal of a spatial coordinate, why would it be under reversal of time? Special relativity has shown the close relationship between spatial and time coordinates. And indeed, it was found that nature is not completely symmetric under time reversal either, even on a microscopic scale.

There is a third symmetry involved in this story of inversion. It involves replacing every particle in a system by the corresponding antiparticle. For every elementary particle, there is a corresponding antiparticle that is its exact opposite. For example, the electron, with electric charge $\vphantom0\raisebox{1.5pt}{$-$}$$e$ and lepton number 1, has an antiparticle, the positron, with charge $e$ and lepton number $\vphantom0\raisebox{1.5pt}{$-$}$1. (Lepton number is a conserved quantity much like charge is.) Bring an electron and a positron together, and they can totally annihilate each other, producing two photons. The net charge was zero, and is still zero. Photons have no charge. The net lepton number was zero, and is still zero. Photons are not leptons and have zero lepton number.

All particles have antiparticles. Protons have antiprotons, neutrons antineutrons, etcetera. Replacing every particle in a system by its antiparticle produces almost the same physics. You can create an antihydrogen atom out of an antiproton and a positron that seems to behave just like a normal hydrogen atom does.

Replacing every particle by its antiparticle is not called particle inversion, as you might think, but “charge conjugation.” That is because physicists recognized that charge inversion would be all wrong; a lot more changes than just the charge. And the particle involved might not even have a charge, like the neutron, with no net charge but a baryon number that inverts, or the neutrino, with no charge but a lepton number that inverts. So physicists figured that if “charge inversion” is wrong anyway, you may as well replace inversion” by “conjugation. That is not the same as inversion, but it was wrong anyway, and conjugation sounds much more sophisticated and it alliterates.

The bottom line is that physics is almost, but not fully, symmetric under spatial inversion, time inversion, and particle inversion. However, physicists currently believe that if you apply all three of these operations together, then the resulting physics is indeed truly the same. There is a theorem called the CPT theorem, (charge, parity, time), that says so under relatively mild assumptions. One way to look at it is to say that systems of antiparticles are the mirror images of systems of normal particles that move backwards in time.

At the time of writing, there is a lot of interest in the possibility that nature may in fact not be exactly the same when the CPT transformations are applied. It is hoped that this may explain why nature ended up consisting almost exclusively of particles, rather than antiparticles.

Symmetry transformations like the ones discussed above form mathematical groups. There are infinitely many different angles that you can rotate a system over or distances that you can translate it over. What is mathematically particularly interesting is how group members combine together into different group members. For example, a rotation followed by another rotation is equivalent to a single rotation over a combined angle. You can even eliminate a rotation by following it by one in the opposite direction. All that is nectar to mathematicians.

The inversion transformations are somewhat different in that they form finite groups. You can either invert or not invert. These finite groups provide much less detailed constraints on the physics. Parity can only be 1 or $\vphantom0\raisebox{1.5pt}{$-$}$1. On the other hand, a component of linear or angular momentum must maintain one specific value out of infinitely many possibilities. But even these constraints remain restricted to the total system. It is the complete system that must maintain the same linear and angular momentum, not the individual parts of it. That reflects that the same rotation angle or translation distance applies for all parts of the system.

Advanced relativistic theories of quantum mechanics postulate symmetries that apply on a local (point by point) basis. A simple example relevant to quantum electrodynamics can be found in addendum {A.19}. Such symmetries narrow down what the physics can do much more because they involve separate parameters at each individual point. Combined with the massive antisymmetrization requirements for fermions, they allow the physics to be deduced in terms of a few remaining numerical parameters. The so-called “standard model” of relativistic quantum mechanics postulates a combination of three symmetries of the form

\begin{displaymath}
{\rm U}(1) \times {\rm SU}(2) \times {\rm SU}(3)
\end{displaymath}

In terms of linear algebra, these are complex matrices that describe rotations of complex vectors in 1, 2, respectively 3 dimensions. The S on the latter two matrices indicates that they are special in the sense that their determinant is 1. The first matrix is characterized by 1 parameter, the angle that the single complex numbers are rotated over. It gives rise to the photon that is the single carrier of the electromagnetic force. The second matrix has 3 parameters, corresponding to the 3 so-called “vector bosons” that are the carriers of the weak nuclear force. The third matrix has 8 parameters, corresponding to the 8 gluons that are the carriers of the strong nuclear force.

There is an entire branch of mathematics, “group theory,” devoted to how group properties relate to the solutions of equations. It is essential to advanced quantum mechanics, but far beyond the scope of this book.


Key Points
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Symmetries of physics give rise to conserved quantities.

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These are of particular interest in obtaining an understanding of complicated and relativistic systems. They can also aid in the solution of simple systems.

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Translational symmetry gives rise to conservation of linear momentum. Rotational symmetry gives rise to conservation of angular momentum.

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Spatial inversion replaces every position vector ${\skew0\vec r}$ by $\vphantom0\raisebox{1.5pt}{$-$}$${\skew0\vec r}$. It produces a conserved quantity called parity.

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There are kinks in the armor of the symmetries under spatial inversion, time reversal, and charge conjugation. However, it is believed that nature is symmetric under the combination of all three.