Sub­sec­tions


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: Ef­fect of a ro­ta­tion of the co­or­di­nate sys­tem on the spher­i­cal co­or­di­nates of a par­ti­cle at an ar­bi­trary lo­ca­tion P.
\begin{figure}\centering
\setlength{\unitlength}{1pt}
\begin{picture}(300,18...
...akebox(0,0){$\phi$}}
\put(41,111){\makebox(0,0){P}}
\end{picture}
\end{figure}

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 $z'$-​axis has been cho­sen along the axis of the de­sired ro­ta­tion. Ro­ta­tion of this co­or­di­nate sys­tem over an an­gle $\gamma$ pro­duces a new co­or­di­nate sys­tem in­di­cated with­out primes. In terms of spher­i­cal co­or­di­nates, the ra­dial po­si­tion $r$ of the par­ti­cle does not change. And nei­ther does the po­lar an­gle $\theta$. But the az­imuthal an­gle $\phi$ does change. As the fig­ure shows, the re­la­tion be­tween the az­imuthal an­gles is

\begin{displaymath}
\phi' = \phi + \gamma
\end{displaymath}

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 $\Psi$ that de­pends on the po­si­tion co­or­di­nates of the par­ti­cles;

\begin{displaymath}
\Psi(r_1,\theta_1,\phi_1,r_2,\theta_2,\phi_2,\ldots ; t)
\end{displaymath}

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 $\Psi$ of the wave func­tion in the ro­tated co­or­di­nate sys­tem are ex­actly the same as the val­ues $\Psi'$ in the orig­i­nal co­or­di­nate sys­tem. But the par­ti­cle co­or­di­nates cor­re­spond­ing to these val­ues do change:

\begin{displaymath}
\Psi(r_1,\theta_1,\phi_1,r_2,\theta_2,\phi_2,\ldots ; t)
=
\Psi'(r_1',\theta_1',\phi_1',r_2',\theta_2',\phi_2',\ldots ; t)
\end{displaymath}

There­fore, con­sid­ered as func­tions, $\Psi'$ and $\Psi$ are dif­fer­ent. How­ever, only the az­imuthal an­gles change. In par­tic­u­lar, putting in the re­la­tion be­tween the az­imuthal an­gles above gives:

\begin{displaymath}
\Psi(r_1,\theta_1,\phi_1,r_2,\theta_2,\phi_2,\ldots ; t)
=...
...,\theta_1,\phi_1+\gamma,r_2,\theta_2,\phi_2+\gamma,\ldots ; t)
\end{displaymath}

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 $z$-​axis.” It will be in­di­cated as ${\cal R}_{z,\gamma}$. What it does is add $\gamma$ to the az­imuthal an­gles of the func­tion. By de­f­i­n­i­tion:

\begin{displaymath}
{\cal R}_{z,\gamma} \Psi'(r_1,\theta_1,\phi_1,r_2,\theta_2,...
...,\theta_1,\phi_1+\gamma,r_2,\theta_2,\phi_2+\gamma,\ldots ; t)
\end{displaymath}

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

\begin{displaymath}
\Psi = {\cal R}_{z,\gamma} \Psi'
\end{displaymath}

Us­ing ${\cal R}_{z,\gamma}$, there is no longer a need for us­ing primes on one set of co­or­di­nates. Take any wave func­tion in terms of the orig­i­nal co­or­di­nates, writ­ten with­out primes. Ap­pli­ca­tion of ${\cal R}_{z,\gamma}$ will turn it into the cor­re­spond­ing wave func­tion in the ro­tated co­or­di­nates, also writ­ten with­out primes.

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 $\Psi$ is a wave func­tion.


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

\begin{displaymath}
\frac{\partial \Psi}{\partial t} = \frac{1}{{\rm i}\hbar} H \Psi
\end{displaymath}

where $H$ is the Hamil­ton­ian. If space has no pre­ferred di­rec­tions, then the Hamil­ton­ian must be the same re­gard­less of an­gu­lar ori­en­ta­tion of the co­or­di­nate sys­tem used.

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 $\gamma$ around the $z$-​axis. If one sys­tem had a dif­fer­ent Hamil­ton­ian than the other, then sys­tems of par­ti­cles would be ob­served to evolve in a dif­fer­ent way in that co­or­di­nate sys­tem. That would pro­vide a fun­da­men­tal dis­tinc­tion be­tween the two co­or­di­nate sys­tem ori­en­ta­tions right there.

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 $x',y',z'$ co­or­di­nate sys­tem is

\begin{displaymath}
H' = - \frac{\hbar^2}{2m_{\rm e}}
\left[
\frac{\partial}{...
...frac{e^2}{4\pi\epsilon_0}\frac{1}{\sqrt{{x'}^2+{y'}^2+{z'}^2}}
\end{displaymath}

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 $x,y,z$ co­or­di­nate sys­tem, sim­ply leave away the primes. The sec­ond term is the po­ten­tial en­ergy in the field of the pro­ton. It is in­versely pro­por­tional to the dis­tance of the elec­tron from the ori­gin. The ex­pres­sion for the dis­tance from the ori­gin is the same in the ro­tated co­or­di­nate sys­tem. Once again, just leave away the primes. The bot­tom line is that you can­not see a dif­fer­ence be­tween the two co­or­di­nate sys­tems by look­ing at their Hamil­to­ni­ans. The ex­pres­sions for the Hamil­to­ni­ans are iden­ti­cal.

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 $x'_{\rm {p}},y'_{\rm {p}},z'_{\rm {p}}$. Its ex­pres­sion too is the same re­gard­less of an­gu­lar ori­en­ta­tion of the co­or­di­nate sys­tem. And in the po­ten­tial en­ergy term, the dis­tance from the ori­gin now be­comes the dis­tance be­tween elec­tron and pro­ton. But the for­mula for the dis­tance be­tween two points is the same re­gard­less of an­gu­lar ori­en­ta­tion of the co­or­di­nate sys­tem. So once again, the ex­pres­sion for the Hamil­ton­ian does not de­pend on the an­gu­lar ori­en­ta­tion of the co­or­di­nate sys­tem.

The equal­ity of the Hamil­to­ni­ans in the orig­i­nal and ro­tated co­or­di­nate sys­tems has a con­se­quence. It leads to a math­e­mat­i­cal re­quire­ment for the op­er­a­tor ${\cal R}_{z,\gamma}$ of the pre­vi­ous sub­sec­tion that de­scribes the ef­fect of a co­or­di­nate sys­tem ro­ta­tion on wave func­tions. This op­er­a­tor must com­mute with the Hamil­ton­ian:

\begin{displaymath}
H {\cal R}_{z,\gamma} = {\cal R}_{z,\gamma} H
\end{displaymath}

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 ${\cal R}_{z,\gamma}$ to get the one in the ro­tated co­or­di­nate sys­tem. Then you can ap­ply the Hamil­ton­ian on that. The other way is to ap­ply the Hamil­ton­ian on the wave func­tion in the orig­i­nal co­or­di­nate sys­tem to find the time de­riv­a­tive in the orig­i­nal co­or­di­nate sys­tem. Then you can use ${\cal R}_{z,\gamma}$ to con­vert that time de­riv­a­tive to the ro­tated sys­tem. The Hamil­ton­ian and ${\cal R}_{z,\gamma}$ get ap­plied in the op­po­site or­der, but the re­sult must still be the same.

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

A sym­me­try of physics is de­scribed by a uni­tary op­er­a­tor that com­mutes with the Hamil­ton­ian.
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.

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 $({\rm i}\Psi)'$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\rm i}(\Psi)'$. But $({\rm i}\Psi)^*$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm i}\Psi^*$. If con­stants come out of an op­er­a­tor as com­plex con­ju­gates, the op­er­a­tor is called “an­ti­lin­ear.” So tak­ing com­plex con­ju­gate is an­ti­lin­ear. An­other is­sue: a lin­ear uni­tary op­er­a­tor pre­serves the in­ner prod­ucts be­tween any two wave func­tions $\Psi_1$ and $\Psi_2$. (That can be ver­i­fied by ex­pand­ing the square mag­ni­tudes of $\Psi_1+\Psi_2$ and $\Psi_1+{\rm i}\Psi_2$). How­ever, tak­ing com­plex con­ju­gate changes in­ner prod­ucts into their com­plex con­ju­gates. Op­er­a­tors that do that are called “an­tiu­ni­tary.” So tak­ing com­plex con­ju­gate is both an­ti­lin­ear and an­tiu­ni­tary. (Of course, in nor­mal lan­guage it is nei­ther. The ap­pro­pri­ate terms would have been con­ju­gate-lin­ear and con­ju­gate-uni­tary. But if you got this far in this book, you know how much chance ap­pro­pri­ate terms have of be­ing used in physics.)

Now the ef­fect of time-re­ver­sal on wave func­tions turns out to be an­ti­lin­ear and an­tiu­ni­tary too, [48, p. 76]. One sim­ple way to think about it is that a straight­for­ward time re­ver­sal would change $e^{-{{\rm i}}Et/\hbar}$ into $e^{{{\rm i}}Et/\hbar}$. Then an ad­di­tional com­plex con­ju­gate will take things back to pos­i­tive en­er­gies. For the same rea­son you do not want to add a com­plex con­ju­gate to spa­tial trans­for­ma­tions or time shifts.


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 $e^{{{\rm i}}Et/\hbar}$ of mag­ni­tude 1 that does not change their phys­i­cal prop­er­ties.

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 $z$-​di­rec­tion means that the net an­gu­lar mo­men­tum of the sys­tem in the $z$-​di­rec­tion, a num­ber, does not change.

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 $e^{{{\rm i}}Et/\hbar}$ changes the eigen­func­tion, but not the eigen­value that would be pro­duced by ap­ply­ing the sym­me­try op­er­a­tor at dif­fer­ent times. The eigen­value can there­fore be looked upon as a spe­cific value of some con­served quan­tity. In those terms, if the state of the sys­tem is given by a dif­fer­ent eigen­func­tion, with a dif­fer­ent eigen­value, it has a dif­fer­ent value for the con­served quan­tity.

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 ${\cal R}_{z,\gamma}$ dis­cussed in the pre­vi­ous sub­sec­tions. You would surely ex­pect a con­served quan­tity of a sys­tem to be a real quan­tity. But the eigen­val­ues of ${\cal R}_{z,\gamma}$ are in gen­eral com­plex num­bers.

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

\begin{displaymath}
e^{{\rm i}\alpha}
\end{displaymath}

where $\alpha$ is some real num­ber. If the eigen­value does not change with time, then nei­ther does $\alpha$, which is ba­si­cally just its log­a­rithm.

But al­though $\alpha$ is real and con­served, still it is not the de­sired con­served quan­tity. Con­sider the pos­si­bil­ity that you per­form an­other ro­ta­tion of the axis sys­tem. Each ro­ta­tion mul­ti­plies the eigen­func­tion by a fac­tor $e^{{\rm i}\alpha}$ for a to­tal of $e^{2{\rm i}\alpha}$. In short, if you dou­ble the an­gle of ro­ta­tion $\gamma$, you also dou­ble the value of $\alpha$. But it does not make sense to say that both $\alpha$ and $2\alpha$ are con­served. If $\alpha$ is con­served, then so is $2\alpha$; that is not a sec­ond con­ser­va­tion law. Since $\alpha$ is pro­por­tional to $\gamma$, it can be writ­ten in the form

\begin{displaymath}
\alpha = m \gamma
\end{displaymath}

where the con­stant of pro­por­tion­al­ity $m$ is in­de­pen­dent of the amount of co­or­di­nate sys­tem ro­ta­tion.

The con­stant $m$ is the de­sired con­served quan­tity. For his­tor­i­cal rea­sons it is called the 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 $L_z$. It turns out that what clas­si­cal physics de­fines as an­gu­lar mo­men­tum is sim­ply a mul­ti­ple of the mag­netic quan­tum num­ber:

\begin{displaymath}
L_z = m \hbar
\end{displaymath}

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 $m$, the eigen­val­ues of ${\cal R}_{z,\gamma}$ are of the form

\begin{displaymath}
e^{{\rm i}m \gamma}
\end{displaymath}

Now if you ro­tate the co­or­di­nate sys­tem over an an­gle $\gamma$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2\pi$, it gets back to the ex­act same po­si­tion as it was in be­fore the ro­ta­tion. The wave func­tion should not change in that case, which means that the eigen­value must be equal to one. And that re­quires that the value of $m$ is an in­te­ger. If $m$ was a frac­tional num­ber, $e^{{{\rm i}}m2\pi}$ would not be 1.

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

\begin{displaymath}
\psi_{nlm}({\skew0\vec r}) = R_{nl}(r) Y_l^m(\theta,\phi)
= R_{nl}(r) \Theta_l^m(\theta) e^{{\rm i}m\phi}
\end{displaymath}

It is the fi­nal ex­po­nen­tial that changes by the ex­pected fac­tor $e^{{{\rm i}}m\gamma}$ when ${\cal R}_{z,\gamma}$ re­places $\phi$ by $\phi+\gamma$.

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

\begin{displaymath}
\psi_{nlm,\rm red}({\skew0\vec r}-{\skew0\vec r}_{\rm p}) \psi_{\rm cg}({\skew0\vec r}_{\rm cg})
\end{displaymath}

The first term is like be­fore, ex­cept that it is com­puted with a re­duced mass that 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 $e^{{{\rm i}}m\gamma}$ when ${\cal R}_{z,\gamma}$ is ap­plied. The sec­ond fac­tor re­flects the mo­tion of the cen­ter of grav­ity of the com­plete atom. If the cen­ter of grav­ity has def­i­nite an­gu­lar mo­men­tum around what­ever point is used as ori­gin, it will pro­duce an ad­di­tional fac­tor $e^{{{\rm i}}m_{\rm {cg}}\gamma}$. (See ad­den­dum {A.6} on how the en­ergy eigen­func­tions $\psi_{\rm {cg}}$ can be writ­ten as spher­i­cal Bessel func­tions of the first kind times spher­i­cal har­mon­ics that have def­i­nite an­gu­lar mo­men­tum. But also see chap­ter 7.9 about the nasty nor­mal­iza­tion is­sues with wave func­tions in in­fi­nite empty space.)

As a fi­nal step, it is de­sir­able to for­mu­late a nicer op­er­a­tor for an­gu­lar mo­men­tum. The ro­ta­tion op­er­a­tors ${\cal R}_{z,\gamma}$ are far from per­fect. One prob­lem is that there are in­fi­nitely many of them, one for every an­gle $\gamma$. And they are all re­lated, a ro­ta­tion over an an­gle $2\gamma$ be­ing the same as two ro­ta­tions over an an­gle $\gamma$.

If you de­fine a ro­ta­tion op­er­a­tor over a very small an­gle, call it ${\cal R}_{z,\varepsilon}$, then you can ap­prox­i­mate any other op­er­a­tor ${\cal R}_{z,\gamma}$ by just ap­ply­ing ${\cal R}_{z,\varepsilon}$ suf­fi­ciently many times. To make this ap­prox­i­ma­tion ex­act, you need to make $\varepsilon$ in­fin­i­tes­i­mally small. But when $\varepsilon$ be­comes zero, ${\cal R}_{z,\varepsilon}$ would be­come just 1. You have lost the nicer op­er­a­tor that you want by go­ing to the ex­treme. The trick to avoid this is to sub­tract the lim­it­ing op­er­a­tor 1, and in ad­di­tion, to avoid that the re­sult­ing op­er­a­tor then be­comes zero, you must also di­vide by $\varepsilon$. The nicer op­er­a­tor is there­fore

\begin{displaymath}
\lim_{\varepsilon\to0}\frac{{\cal R}_{z,\varepsilon} - 1}{\varepsilon}
\end{displaymath}

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

\begin{displaymath}
\lim_{\varepsilon\to0}\frac{{\cal R}_{z,\varepsilon} - 1}{\...
...i(r,\theta,\phi+\varepsilon)-\Psi(r,\theta,\phi)}{\varepsilon}
\end{displaymath}

By de­f­i­n­i­tion, the fi­nal term is the par­tial de­riv­a­tive of $\Psi$ with re­spect to $\phi$. So the new op­er­a­tor is just the op­er­a­tor $\partial$$\raisebox{.5pt}{$/$}$$\partial\phi$!

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

\begin{displaymath}
\lim_{\varepsilon\to0}\frac{e^{{\rm i}m\varepsilon} - 1}{\varepsilon}
= {\rm i}m
\end{displaymath}

If you add a fac­tor $\hbar$$\raisebox{.5pt}{$/$}$${\rm i}$ to the op­er­a­tor, the eigen­val­ues of the op­er­a­tor are go­ing to be $m\hbar$, the quan­tity de­fined in clas­si­cal physics as the an­gu­lar mo­men­tum. So you are led to de­fine the an­gu­lar mo­men­tum op­er­a­tor of a sin­gle par­ti­cle as:

\begin{displaymath}
\L _z \equiv \frac{\hbar}{{\rm i}} \frac{\partial}{\partial\phi}
\end{displaymath}

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:

\begin{displaymath}
\fbox{$\displaystyle
\L _z \equiv
\frac{\hbar}{{\rm i}}
...
...lon\to0}\frac{{\cal R}_{z,\varepsilon} - 1}{\varepsilon}
$} %
\end{displaymath} (A.76)

The sys­tem has def­i­nite an­gu­lar mo­men­tum $m\hbar$ if

\begin{displaymath}
\L _z \Psi = m \hbar \Psi
\end{displaymath}

Con­sider now what hap­pens if the an­gu­lar op­er­a­tor $\L _z$ as de­fined above is ap­plied to the wave func­tion of a sys­tem of mul­ti­ple par­ti­cles, still with­out spin. It pro­duces

\begin{displaymath}
\L _z \Psi = \frac{\hbar}{{\rm i}}
\lim_{\varepsilon\to0} ...
..._1,\theta_1,\phi_1,r_2,\theta_2,\phi_2,\ldots)}
{\varepsilon}
\end{displaymath}

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

\begin{displaymath}
\L _z \Psi = \frac{\hbar}{{\rm i}}
\left[
\frac{\partial...
..._1} +
\frac{\partial}{\partial\phi_2} +
\ldots
\right] \Psi
\end{displaymath}

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

\begin{displaymath}
\L _z \Psi = \left[\L _{z,1} + \L _{z,2} + \ldots\right] \Psi
\end{displaymath}

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 $i$ has def­i­nite an­gu­lar mo­men­tum $m_i\hbar$ if

\begin{displaymath}
\L _{z,i} \Psi \equiv \frac{\hbar}{{\rm i}} \frac{\partial}{\partial\phi_i} \Psi
= m_i \hbar \Psi
\end{displaymath}

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 ${\cal R}_{z,\gamma}$ can be iden­ti­fied in terms of it. It turns out to be

\begin{displaymath}
\fbox{$\displaystyle
{\cal R}_{z,\gamma}=\exp\left(\frac{{\rm i}}{\hbar}\L _z\gamma\right)
$}
\end{displaymath} (A.77)

To check that it does in­deed take the form above, ex­pand the ex­po­nen­tial in a Tay­lor se­ries. Then ap­ply it on an eigen­func­tion with an­gu­lar mo­men­tum $L_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m\hbar$. The ef­fect is seen to be to mul­ti­ply the eigen­func­tion by the Tay­lor se­ries of $e^{{\rm i}{m}\gamma}$ as it should. So ${\cal R}_{z,\gamma}$ as given above gets all eigen­func­tions right. It must there­fore be cor­rect since the eigen­func­tions are com­plete.

Now con­sider the gen­er­a­tor of ro­ta­tions in terms of the in­di­vid­ual par­ti­cles. Since $\L _z$ is the sum of the an­gu­lar mo­menta of the in­di­vid­ual par­ti­cles,

\begin{displaymath}
{\cal R}_{z,\gamma} =
\exp\left(\frac{{\rm i}}{\hbar}\L _{...
... \exp\left(\frac{{\rm i}}{\hbar}\L _{z,2}\gamma\right)
\ldots
\end{displaymath}

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 $i$ has def­i­nite an­gu­lar mo­men­tum $m_i\hbar$, then it con­tributes a fac­tor $e^{{{\rm i}}m_i\gamma}$ to ${\cal R}_{z,\gamma}$.

How about spin? The nor­mal an­gu­lar mo­men­tum dis­cussed so far sug­gests its true mean­ing. If a par­ti­cle $i$ has def­i­nite spin an­gu­lar mo­men­tum in the $z$-​di­rec­tion $m_{s,i}\hbar$, then pre­sum­ably the wave func­tion changes by an ad­di­tional fac­tor $e^{{{\rm i}}m_{s,i}\gamma}$ when you ro­tate the axis sys­tem over an an­gle $\gamma$.

But there is some­thing cu­ri­ous here. If the axis sys­tem is ro­tated over an an­gle $2\pi$, it is back in its orig­i­nal po­si­tion. So you would ex­pect that the wave func­tion is also again the same as be­fore the ro­ta­tion. And if there is just or­bital an­gu­lar mo­men­tum, then that is in­deed the case, be­cause $e^{{{\rm i}}m2\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 as long as $m$ is an in­te­ger, (2.5). But for fermi­ons the spin an­gu­lar mo­men­tum $m_s$ in a given di­rec­tion is half-in­te­ger, and $e^{{\rm i}\pi}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom{0}\raisebox{1.5pt}{$-$}$1. There­fore the wave func­tion of a fermion changes sign when the co­or­di­nate sys­tem is ro­tated over $2\pi$ and is back in its orig­i­nal po­si­tion. That is true even if there is un­cer­tainty in the spin an­gu­lar mo­men­tum. For ex­am­ple, the wave func­tion of a fermion with spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ can be writ­ten as, chap­ter 5.5.1,

\begin{displaymath}
\Psi_+{\uparrow}+\Psi_-{\downarrow}
\end{displaymath}

where the first term has $\frac12\hbar$ an­gu­lar mo­men­tum in the $z$-​di­rec­tion and the sec­ond term $-\frac12\hbar$. Each term changes sign un­der a turn of the co­or­di­nate sys­tem by $2\pi$. So the com­plete wave func­tion changes sign. More gen­er­ally, for a sys­tem with an odd num­ber of fermi­ons the wave func­tion changes sign when the co­or­di­nate sys­tem is ro­tated over $2\pi$. For a sys­tem with an even num­ber of fermi­ons, the wave func­tion re­turns to the orig­i­nal value.

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 $4\pi$ in­stead of $2\pi$. (How­ever, it may be only a math­e­mat­i­cal ar­ti­fact. The an­ti­sym­metriza­tion re­quire­ment im­plies that the true sys­tem in­cludes all elec­trons in the uni­verse. Pre­sum­ably, the num­ber of fermi­ons in the uni­verse is in­fi­nite. That makes the ques­tion whether the num­ber is odd or even unan­swer­able. If the num­ber of fermi­ons does turn out to be fi­nite, this book will re­con­sider the ques­tion when peo­ple fin­ish count­ing.)

(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 ${\cal T}_{z,d}$ shifts the co­or­di­nate sys­tem over a dis­tance $d$ in the $z$-​di­rec­tion. Its eigen­val­ues are of the form

\begin{displaymath}
e^{{\rm i}k_z d}
\end{displaymath}

where $k_z$ is a real num­ber, in­de­pen­dent of the amount of trans­la­tion $d$, that is called the wave num­ber. Fol­low­ing the same ar­gu­ments as for an­gu­lar mo­men­tum, $k_z$ is a pre­served quan­tity. In clas­si­cal physics not $k_z$, but $p_z$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\hbar}k_z$ is de­fined as the con­served quan­tity. To get the op­er­a­tor for this quan­tity, form the op­er­a­tor
\begin{displaymath}
\fbox{$\displaystyle
{\widehat p}_z = \frac{\hbar}{{\rm i}...
...lon\to0}\frac{{\cal T}_{z,\varepsilon} - 1}{\varepsilon}
$} %
\end{displaymath} (A.78)

For a sin­gle par­ti­cle, this be­comes the usual lin­ear mo­men­tum op­er­a­tor $\hbar\partial$$\raisebox{.5pt}{$/$}$${\rm i}\partial{z}$. For mul­ti­ple par­ti­cles, the lin­ear mo­menta add up.

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

\begin{displaymath}
\psi_{nlm}({\skew0\vec r})
\end{displaymath}

with ${\skew0\vec r}$ the po­si­tion of the elec­tron. Shift­ing the co­or­di­nate sys­tem for this so­lu­tion means re­plac­ing ${\skew0\vec r}$ by ${\skew0\vec r}+d{\hat k}$. That shifts the po­si­tion of the elec­tron with­out chang­ing the po­si­tion of the pro­ton. The physics is not the same af­ter such a shift. Cor­re­spond­ingly, the eigen­func­tions do not change by a fac­tor of the form $e^{{{\rm i}}k_zd}$ un­der the shift. Just look­ing at the ground state,

\begin{displaymath}
\psi_{100}({\skew0\vec r}) = \frac{1}{\sqrt{\pi a_0^3}} e^{-\vert{\skew0\vec r}\vert/a_0}
\end{displaymath}

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

\begin{displaymath}
\psi_{nlm,\rm red}({\skew0\vec r}-{\skew0\vec r}_{\rm p}) e^{{\rm i}{\vec k}\cdot{\skew0\vec r}_{\rm cg}}
\end{displaymath}

where ${\vec k}$ is a con­stant wave num­ber vec­tor. The first fac­tor does not change un­der co­or­di­nate shifts be­cause the vec­tor ${\skew0\vec r}-{\skew0\vec r}_{\rm {p}}$ from pro­ton to elec­tron does not. The ex­po­nen­tial changes by the ex­pected fac­tor $e^{{{\rm i}}k_zd}$ be­cause the po­si­tion ${\skew0\vec r}_{\rm {cg}}$ of the cen­ter of grav­ity of the atom changes by an amount $d$ in the $z$-​di­rec­tion.

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 $d$. (This as­sumes that the $z$-​axis is cho­sen along one of the prim­i­tive vec­tors of the crys­tal struc­ture.) The eigen­val­ues are still of the form $e^{{{\rm i}}k_zd}$. How­ever, un­like for lin­ear mo­men­tum, the trans­la­tion $d$ must be the crys­tal pe­riod, or an in­te­ger mul­ti­ple of it. There­fore, the op­er­a­tor ${\widehat p}_z$ is not use­ful; the sym­me­try does not con­tinue to ap­ply in the limit $d\to0$.

The con­served quan­tity in this case is just the $e^{{{\rm i}}k_zd}$ eigen­value of ${\cal T}_{z,d}$. It is not pos­si­ble from that eigen­value to uniquely de­ter­mine a value of $k_z$ and the cor­re­spond­ing crys­tal mo­men­tum ${\hbar}k_z$. Val­ues of $k_z$ that dif­fer by a whole mul­ti­ple of $2\pi$$\raisebox{.5pt}{$/$}$$d$ pro­duce the same eigen­value. But Bloch waves have the same in­de­ter­mi­nacy in their value of $k_z$ any­way. In fact, Bloch waves are eigen­func­tions of ${\cal T}_{z,d}$ as well as en­ergy eigen­func­tions.

One con­se­quence of the in­de­ter­mi­nacy in $k_z$ is an in­creased num­ber of pos­si­ble elec­tro­mag­netic tran­si­tions. Typ­i­cal elec­tro­mag­netic ra­di­a­tion has a wave length that is large com­pared to the atomic spac­ing. Es­sen­tially the elec­tro­mag­netic field is the same from one atom to the next. That means that it has neg­li­gi­ble crys­tal mo­men­tum, us­ing the small­est of the pos­si­ble val­ues of $k_x$ as mea­sure. There­fore the ra­di­a­tion can­not change the con­served eigen­value $e^{{{\rm i}}k_zd}$. But it can still pro­duce elec­tron tran­si­tions be­tween two Bloch waves that have been as­signed dif­fer­ent $k_z$ val­ues in some ex­tended zone scheme, chap­ter 6.22.4. As long as the two $k_z$ val­ues dif­fer by a whole mul­ti­ple of $2\pi$$\raisebox{.5pt}{$/$}$$d$, the ac­tual eigen­value $e^{{{\rm i}}k_zd}$ does not change. In that case there is no vi­o­la­tion of the con­ser­va­tion law in the tran­si­tion. The am­bi­gu­ity in $k_z$ val­ues may be elim­i­nated by switch­ing to a re­duced zone scheme de­scrip­tion, chap­ter 6.22.4.

The time shift op­er­a­tor ${\cal U}_\tau$ shifts the time co­or­di­nate over an in­ter­val $\tau$. In empty space, its eigen­func­tions are ex­actly the en­ergy eigen­func­tions. Its eigen­val­ues are of the form

\begin{displaymath}
e^{-{\rm i}\omega\tau}
\end{displaymath}

where clas­si­cal physics de­fines $\hbar\omega$ as the en­ergy $E$. The en­ergy op­er­a­tor can be de­fined cor­re­spond­ingly, and is sim­ply the Hamil­ton­ian:
\begin{displaymath}
H = {\rm i}\hbar \lim_{\varepsilon\to0}\frac{{\cal U}_\vare...
... - 1}{\varepsilon}
= {\rm i}\hbar \frac{\partial}{\partial t}
\end{displaymath} (A.79)

In other words, we have rea­soned in a cir­cle and red­erived the Schrö­din­ger equa­tion from time shift sym­me­try. But you could gen­er­al­ize the rea­son­ing to the mo­tion of par­ti­cles in an ex­ter­nal field that varies pe­ri­od­i­cally in time.

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 180$\POW9,{\circ}$ around the axis nor­mal to the mir­ror.

(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 $x$ by $\vphantom{0}\raisebox{1.5pt}{$-$}$$x$ in some suit­ably cho­sen $xy$-​co­or­di­nate sys­tem. Sub­se­quently re­plac­ing $y$ by $\vphantom{0}\raisebox{1.5pt}{$-$}$$y$ would amount to a sec­ond mir­ror­ing that would re­store a non­mir­ror im­age. In those terms, in three di­men­sions it is re­plac­ing $z$ by $\vphantom{0}\raisebox{1.5pt}{$-$}$$z$ that pro­duces the fi­nal mir­ror im­age in spa­tial in­ver­sion.)

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 $\vphantom{0}\raisebox{1.5pt}{$-$}$${\skew0\vec r}$ back into the orig­i­nal ${\skew0\vec r}$. Then the wave func­tion is again the same. In other words, ap­ply­ing spa­tial in­ver­sion twice mul­ti­plies wave func­tions by 1. It fol­lows that the square of every eigen­value is 1. And if the square of an eigen­val­ues is 1, then the eigen­value it­self must be ei­ther 1 or $\vphantom{0}\raisebox{1.5pt}{$-$}$1. In the same no­ta­tion as used for an­gu­lar mo­men­tum, the eigen­val­ues of the spa­tial in­ver­sion op­er­a­tor can there­fore be writ­ten as

\begin{displaymath}
e^{{\rm i}m'\pi} = (-1)^{m'}
\end{displaymath} (A.80)

where $m'$ must be in­te­ger. How­ever, it is point­less to give an ac­tual value for $m'$; the only thing that makes a dif­fer­ence for the eigen­value is whether $m'$ is even or odd. There­fore, par­ity is sim­ply called odd” or “mi­nus one or neg­a­tive if the eigen­value is $\vphantom{0}\raisebox{1.5pt}{$-$}$1, and even” or “one or pos­i­tive if the eigen­value is 1.

In a sys­tem, the $\pm1$ par­ity eigen­val­ues of the in­di­vid­ual par­ti­cles mul­ti­ply to­gether. That is just like how the eigen­val­ues of the gen­er­a­tor of ro­ta­tion ${\cal R}_{z,\gamma}$ mul­ti­ply to­gether for an­gu­lar mo­men­tum. Any par­ti­cle with even par­ity has no ef­fect on the sys­tem par­ity; it mul­ti­ples the to­tal eigen­value by 1. On the other hand, each par­ti­cle with odd par­ity flips over the to­tal par­ity from odd to even or vice-versa; it mul­ti­plies the to­tal eigen­value by $\vphantom{0}\raisebox{1.5pt}{$-$}$1. Par­ti­cles can also have in­trin­sic par­ity. How­ever, there is no half-in­te­ger par­ity like there is half-in­te­ger spin.


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 $\skew3\vec{\cal E}$. Ba­sic physics says that the elec­tric field is mi­nus the de­riv­a­tive of a po­ten­tial $\varphi$. The po­ten­tial $\varphi$ is com­monly known as the volt­age in elec­tri­cal ap­pli­ca­tions. Now it too has a sym­me­try: adding some ar­bi­trary con­stant, call it $C$, to $\varphi$ does not make a dif­fer­ence. Only dif­fer­ences in volt­age can be ob­served phys­i­cally. That is a very sim­ple ex­am­ple of a gauge sym­me­try, a sym­me­try in an un­ob­serv­able field, here the po­ten­tial $\varphi$.

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 $\varphi$ and charge con­ser­va­tion. Sup­pose that, say, a few pho­tons cre­ate an elec­tron and an an­ti­neu­trino. That can sat­isfy con­ser­va­tion of an­gu­lar mo­men­tum and of lep­ton num­ber, but it would vi­o­late charge con­ser­va­tion. Pho­tons have no charge, and nei­ther have neu­tri­nos. So the neg­a­tive charge $\vphantom{0}\raisebox{1.5pt}{$-$}$$e$ of the elec­tron would ap­pear out of noth­ing. But so what? Pho­tons can cre­ate elec­tron-positron pairs, so why not elec­tron-an­ti­neu­trino pairs?

The prob­lem is that in elec­tro­sta­t­ics an elec­tron has an elec­tro­sta­tic en­ergy $-e\varphi$. There­fore the pho­tons would need to pro­vide not just the rest mass and ki­netic en­ergy for the elec­tron and an­ti­neu­trino, but also an ad­di­tional elec­tro­sta­tic en­ergy $-e\varphi$. That ad­di­tional en­ergy could be de­ter­mined from com­par­ing the en­ergy of the pho­tons against that of the elec­tron-an­ti­neu­trino pair. And that would mean that the value of $\varphi$ at the point of pair cre­ation has been de­ter­mined. Not just a dif­fer­ence in $\varphi$ val­ues be­tween dif­fer­ent points. And that would mean that the value of the con­stant $C$ would be fixed. So na­ture would not re­ally have the gauge sym­me­try that a con­stant in the po­ten­tial is ar­bi­trary.

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 $+e$ of the positron makes the net charge that is cre­ated zero. Sim­i­larly, in beta de­cay, an un­charged neu­tron cre­ates an elec­tron-an­ti­neu­trino pair with charge $\vphantom{0}\raisebox{1.5pt}{$-$}$$e$, but also a pro­ton with charge $+e$.

You might of course won­der whether an elec­tro­sta­tic en­ergy con­tri­bu­tion $-e\varphi$ is re­ally needed to cre­ate an elec­tron. It is be­cause of en­ergy con­ser­va­tion. Oth­er­wise there would be a prob­lem if an elec­tron-an­ti­neu­trino pair was cre­ated at a lo­ca­tion P and dis­in­te­grated again at a dif­fer­ent lo­ca­tion Q. The elec­tron would pick up a ki­netic en­ergy $-e(\varphi_{\rm {P}}-\varphi_{\rm {Q}})$ while trav­el­ing from P to Q. With­out elec­tro­sta­tic con­tri­bu­tions to the elec­tron cre­ation and an­ni­hi­la­tion en­er­gies, that ki­netic en­ergy would make the pho­tons pro­duced by the pair an­ni­hi­la­tion more en­er­getic than those de­stroyed in the pair cre­ation. So the com­plete process would cre­ate ad­di­tional pho­ton en­ergy out of noth­ing.

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 $\skew3\vec{\cal E}$ but also a mag­netic field $\skew2\vec{\cal B}$. There is a cor­re­spond­ing ad­di­tional so-called vec­tor po­ten­tial $\skew3\vec A$ in ad­di­tion to the scalar po­ten­tial $\varphi$. The re­la­tion be­tween these po­ten­tials and the elec­tric and mag­netic fields is given by, chap­ter 13.1:

\begin{displaymath}
\skew3\vec{\cal E}= -\nabla \varphi - \frac{\partial \skew3...
...ial t}
\qquad
\skew2\vec{\cal B}= \nabla \times \skew3\vec A
\end{displaymath}

Here $\nabla$, nabla, is the dif­fer­en­tial op­er­a­tor of vec­tor cal­cu­lus (cal­cu­lus III in the U.S. sys­tem):

\begin{displaymath}
\nabla \equiv
{\hat\imath}\frac{\partial}{\partial x} +
{...
...c{\partial}{\partial y} +
{\hat k}\frac{\partial}{\partial z}
\end{displaymath}

The gauge prop­erty now be­comes more gen­eral. The con­stant $C$ that can be added to $\varphi$ in elec­tro­sta­t­ics no longer needs to be con­stant. In­stead, it can be taken to be the time-de­riv­a­tive of any ar­bi­trary func­tion $\chi(x,y,z,t)$. How­ever, the gra­di­ent of this func­tion must also be sub­tracted from $\skew3\vec A$. In par­tic­u­lar, the po­ten­tials

\begin{displaymath}
\varphi' = \varphi + \frac{\partial \chi}{\partial t}
\qquad
\skew3\vec A' = \skew3\vec A- \nabla \chi
\end{displaymath}

pro­duce the ex­act same elec­tric and mag­netic fields as $\varphi$ and $\skew3\vec A$. So they are phys­i­cally equiv­a­lent. They pro­duce the same ob­serv­able mo­tion.

How­ever, the wave func­tion com­puted us­ing the po­ten­tials $\varphi'$ and $\skew3\vec A'$ is dif­fer­ent from the one com­puted us­ing $\varphi$ and $\skew3\vec A$. The rea­son is that the Hamil­ton­ian uses the po­ten­tials rather than the elec­tric and mag­netic fields. Ig­nor­ing spin, the Hamil­ton­ian of an elec­tron in an elec­tro­mag­netic field is, chap­ter 13.1:

\begin{displaymath}
H =
\frac{1}{2m_{\rm e}}\left(\frac{\hbar}{{\rm i}}\nabla + e \skew3\vec A\right)^2 - e \varphi
\end{displaymath}

It can be seen by crunch­ing it out that if $\Psi$ sat­is­fies the Schrö­din­ger equa­tion in which the Hamil­ton­ian is formed with $\varphi$ and $\skew3\vec A$, then
\begin{displaymath}
\Psi' = e^{{\rm i}e \chi/\hbar} \Psi
\end{displaymath} (A.81)

sat­is­fies the one in which $H$ is formed with $\varphi'$ and $\skew3\vec A'$.

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 $\vert\Psi\vert^2$ de­ter­mines the prob­a­bil­ity per unit vol­ume of find­ing the elec­tron at a given lo­ca­tion. But the wave func­tion is a com­plex num­ber; it can al­ways be writ­ten in the form

\begin{displaymath}
\Psi = e^{{\rm i}\alpha}\vert\Psi\vert
\end{displaymath}

where $\alpha$ is a real quan­tity cor­re­spond­ing to a phase an­gle. This an­gle is not di­rectly ob­serv­able; it drops out of the mag­ni­tude of the wave func­tion. And the gauge prop­erty above shows that not only is $\alpha$ not ob­serv­able, it can be any­thing. For, the func­tion $\chi$ can change $\alpha$ by a com­pletely ar­bi­trary amount $e\chi$$\raisebox{.5pt}{$/$}$$\hbar$ and it re­mains a so­lu­tion of the Schrö­din­ger equa­tion. The only vari­ables that change are the equally un­ob­serv­able po­ten­tials $\varphi$ and $\skew3\vec A$.

As noted ear­lier, a sym­me­try means that you can do some­thing and it does not make a dif­fer­ence. Since $\alpha$ can be cho­sen com­pletely ar­bi­trary, vary­ing with both lo­ca­tion and time, this is a very strong sym­me­try. Zee writes, (Quan­tum Field The­ory in a Nut­shell, 2003, p. 135): "The mod­ern phi­los­o­phy is to look at [the equa­tions of quan­tum elec­tro­dy­nam­ics] as a re­sult of [the gauge sym­me­try above]. If we want to con­struct a gauge-in­vari­ant rel­a­tivis­tic field the­ory in­volv­ing a spin $\leavevmode \kern.03em\raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em
/\kern-.21em\lower.56ex\hbox{\the\scriptfont0 2}\kern.05em$ and a spin 1 field, then we are forced to quan­tum elec­tro­dy­nam­ics."

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 $e^{{{\rm i}}e\chi/\hbar}$ cor­re­sponds to ro­tat­ing it over an an­gle $e\chi$$\raisebox{.5pt}{$/$}$$\hbar$ around the ori­gin in that plane. In those terms, the wave func­tion can be ro­tated over an ar­bi­trary, vary­ing, an­gle in the com­plex plane and it still sat­is­fies the Schrö­din­ger equa­tion.

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 $F^{\mu\nu}$ are de­riv­a­tives of this wave func­tion, [24, p. 239].


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.