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 $\vphantom{0}\raisebox{1.5pt}{$-$}$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 $\vphantom{0}\raisebox{1.5pt}{$-$}$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 $\vphantom{0}\raisebox{1.5pt}{$-$}$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 $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ 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 $\vphantom{0}\raisebox{1.5pt}{$-$}$${\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 $\vphantom{0}\raisebox{1.5pt}{$-$}$$e$ and lep­ton num­ber 1, has an an­tipar­ti­cle, the positron, with charge $e$ and lep­ton num­ber $\vphantom{0}\raisebox{1.5pt}{$-$}$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 $\vphantom{0}\raisebox{1.5pt}{$-$}$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

$${\rm U}(1) \times {\rm SU}(2) \times {\rm SU}(3)$$

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 $\vphantom{0}\raisebox{1.5pt}{$-$}$${\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.