Celebratio Mathematica

Andrew Mattei Gleason

Andy Gleason and quantum mechanics

by Paul R. Chernoff

About Andy

I met Andy at the be­gin­ning of my second year at Har­vard when I signed up for his gradu­ate ana­lys­is course. Andy briefly in­ter­viewed pro­spect­ive stu­dents to see if they had enough back­ground to be­ne­fit from this rather soph­ist­ic­ated course. I told Andy that I owned a num­ber of ad­vanced books which I hadn’t read.

The course was both a chal­lenge and a pleas­ure. I can only echo what oth­ers have said about Andy’s lu­min­ous clar­ity and massive ab­stract power. But I must ad­mit that the lec­tures, al­ways ex­cit­ing, wer­en’t ab­so­lutely per­fect; in the course of a year Andy made one genu­ine blun­der. As to his fam­ous speed, John Schwarz, the well-known string the­or­ist, once said after class that Andy had “the meta­bol­ism of a hum­ming­bird”.

I was ex­tremely lucky that Andy was af­fil­i­ated with Low­ell House, my un­der­gradu­ate res­id­ence. Every week Andy came for lunch, where we sat around a large cir­cu­lar table. That’s how Andy and I be­came friends. Of course we dis­cussed a lot of math­em­at­ics around that table, but lots of oth­er things, in­clud­ing Andy’s “war stor­ies”. I am not sur­prised that someone kept a great treas­ure: all of Andy’s nap­kin manuscripts.

Al­most any math­em­at­ic­al prob­lem could in­trigue Andy. At one of the an­nu­al math de­part­ment pic­nics, he had fun fig­ur­ing out how to do cube roots on an aba­cus. But most im­port­ant was his un­pre­ten­tious­ness, open­ness, and great in­terest in stu­dents. I sup­pose that all teach­ers are im­pa­tient at times; no doubt Andy was sorely tried on oc­ca­sion. But he rarely, if ever, showed it. The stu­dents in one of his classes gave him a framed copy of Pi­cas­so’s early paint­ing Moth­er and Child. Per­haps they chose this gift to sym­bol­ize Andy’s nur­tur­ing of them. It’s re­gret­table that there are some teach­ers for whom Guer­nica would be more ap­pro­pri­ate.

Quantum mechanics

Gleason with koala, Cleland Wildlife Park, Adelaide, Australia, 1988.
Photo courtesy of Jean Berko Gleason.

In this sec­tion we set the stage for a dis­cus­sion of Andy’s unique con­tri­bu­tion to phys­ics: his re­mark­able pa­per “Meas­ures on the closed sub­spaces of a Hil­bert space” [1]. It’s in­ter­est­ing in sev­er­al ways: its his­tory; its in­flu­ence in math­em­at­ics; and es­pe­cially its un­ex­pec­ted im­port­ance to the ana­lys­is of “hid­den vari­able” the­or­ies of quantum mech­an­ics by the phys­i­cist John Bell.

In clas­sic­al mech­an­ics, the state of a particle of mass \( m \) is giv­en by its po­s­i­tion and mo­mentum. The mo­tion or dy­nam­ics of a set of particles with as­so­ci­ated forces is de­term­ined by New­ton’s second law of mo­tion, a sys­tem of or­din­ary dif­fer­en­tial equa­tions. This yields a pic­ture of the mac­ro­scop­ic world which matches our in­tu­ition. The ul­tra­mic­ro­scop­ic world re­quires a quite dif­fer­ent de­scrip­tion. The state of a particle of mass \( m \) in \( \mathbb{R}^3 \) is a com­plex val­ued func­tion \( \psi \) on \( \mathbb{R}^3 \). Its mo­mentum is sim­il­arly de­scribed by the func­tion \( \phi \), the Four­i­er trans­form of \( \psi \), nor­mal­ized by the pres­ence in the ex­po­nent of the ra­tio \( \frac{h}{m} \), where \( h \) is Planck’s con­stant. (Us­ing stand­ard prop­er­ties of the Four­i­er trans­form, one can de­duce Heis­en­berg’s un­cer­tainty prin­ciple.) For \( n \) particles, \( \psi \) is defined on \( \mathbb{R}^{3n} \). This is a bril­liant ex­tra­pol­a­tion of the ini­tial ideas of DeBroglie. The Schrödinger equa­tion de­term­ines the dy­nam­ics. If both \( \psi \) and \( \phi \) are largely con­cen­trated around \( n \) points in po­s­i­tion and mo­mentum space re­spect­ively, then the quantum state re­sembles a blurry pic­ture of the clas­sic­al state. The more massive the particles, the less the blur­ri­ness (pro­tons versus base­balls).

The fun­da­ment­al in­ter­pret­a­tion of the “wave func­tion” \( \psi \) is the work of Max Born.1 His pa­per ana­lyz­ing col­li­sions of particles ends with the con­clu­sion that \( |\psi |^2 \) should be in­ter­preted as the prob­ab­il­ity dis­tri­bu­tion for the po­s­i­tions of the particles. There­fore the wave func­tion must be a unit vec­tor in \( L^2 \). Thus did Hil­bert space enter quantum mech­an­ics.

Pri­or to Schrödinger’s wave mech­an­ics, Heis­en­berg had be­gun to de­vel­op a the­ory in which ob­serv­able quant­it­ies are rep­res­en­ted by Her­mitian-sym­met­ric in­fin­ite square ar­rays. He de­vised a “pe­cu­li­ar” law for mul­tiply­ing two ar­rays by an in­geni­ous use of the phys­ic­al mean­ing of their entries. Born had learned mat­rix the­ory when he was a stu­dent and real­ized (after a week of “agony”) that Heis­en­berg’s re­cipe was just mat­rix mul­ti­plic­a­tion. Hence the Heis­en­berg the­ory is called mat­rix mech­an­ics. (Schrödinger showed that mat­rix mech­an­ics and wave mech­an­ics are math­em­at­ic­ally equi­val­ent.) As in clas­sic­al mech­an­ics, the dy­nam­ics of a quantum sys­tem is de­term­ined from its en­ergy \( H \). Peri­od­ic or­bits cor­res­pond to the ei­gen­val­ues of \( H \), i.e., the dis­crete en­ergy levels. The cal­cu­la­tion of the ei­gen­val­ues is very dif­fi­cult, save for a few simple sys­tems. The en­ergy levels for the hy­dro­gen atom were in­geni­ously cal­cu­lated by Wolfgang Pauli; his res­ults agreed with Bo­hr’s cal­cu­la­tions done at the very be­gin­ning of the “old” quantum the­ory.

Born was quite fa­mil­i­ar with Hil­bert’s the­ory of in­teg­ral equa­tions in \( L^2 \). Ac­cord­ingly, he was able to in­ter­pret Heis­en­berg’s matrices as Her­mitian sym­met­ric ker­nels with re­spect to some or­thonor­mal basis, which might just as well be re­garded as the cor­res­pond­ing in­teg­ral op­er­at­ors on \( L^2 \). Form­ally, every Her­mitian mat­rix could be re­garded as an in­teg­ral op­er­at­or, usu­ally with a very sin­gu­lar ker­nel. (The most fa­mil­i­ar ex­ample is the iden­tity, with ker­nel the Dir­ac delta func­tion.) In this way, Born ini­ti­ated the stand­ard pic­ture of ob­serv­ables as Her­mitian op­er­at­ors A on \( L^2 \). But at that time, the phys­i­cists did not grasp the im­port­ant dis­tinc­tion between un­boun­ded Her­mitian op­er­at­ors and un­boun­ded self-ad­joint op­er­at­ors. That was greatly cla­ri­fied by John von Neu­mann, ma­jor de­veloper of the the­ory of un­boun­ded self-ad­joint op­er­at­ors.

Hav­ing in­ter­preted \( |\psi |^2 \) as the prob­ab­il­ity dis­tri­bu­tion for the po­s­i­tions of particles, Born went on to de­vise what im­me­di­ately be­came the stand­ard in­ter­pret­a­tion of meas­ure­ments in quantum mech­an­ics: the prob­ab­il­ity that a meas­ure­ment of a quantum sys­tem will yield a par­tic­u­lar res­ult.

Born’s line of thought was this. A state of a quantum sys­tem cor­res­ponds to a unit vec­tor \( \psi \in L^2 \). What are the pos­sible val­ues of a meas­ure­ment of the ob­serv­able rep­res­en­ted by the op­er­at­or A, and what is the prob­ab­il­ity that a spe­cif­ic value is ob­served? Born dealt only with op­er­at­ors with a dis­crete spec­trum, namely, the set of all its ei­gen­val­ues. For sim­pli­city, as­sume that there are no mul­tiple ei­gen­val­ues. Let \( \phi_n \) be the unit ei­gen­vector with ei­gen­value \( \lambda_n. \) These form an or­thonor­mal basis of \( L^2 \). Ex­pand \( \psi \) as a series \( \sum_k c_k \phi_k \). Since \( \|\psi \|^2 = 1 \), we get \( \sum_k |c_k |^2 = 1 \). Born’s in­sight was that any meas­ure­ment must yield one of the ei­gen­val­ues \( \lambda_n \) of \( A \), and \( |c_n |^2 \) is the prob­ab­il­ity that the res­ult of the meas­ure­ment is \( \lambda_n \). This is known as Born’s rule. It fol­lows that the ex­pec­ted value of a meas­ure­ment of \( A \) is \( \sum_k |c_k |^2 \lambda_k \). Note that this sum equals the in­ner product \( (A\psi , \psi) \). This is the same as \( \operatorname{trace}(P A) \), where \( P \) is the pro­jec­tion onto the one-di­men­sion­al sub­space spanned by \( \psi \). (To jump ahead, George Mackey wondered if Born’s rule might in­volve some ar­bit­rary choices. Gleason ruled this out.)

John von Neu­mann was the cre­at­or of the ab­stract the­ory of quantum mech­an­ics. In his the­ory, a pure state is a unit vec­tor in a Hil­bert space \( \mathcal{H} \) . Ob­serv­ables are self-ad­joint op­er­at­ors, un­boun­ded in gen­er­al, whose spec­trum may be any Borel sub­set of \( \mathbb{R} \). Von Neu­mann also de­veloped the im­port­ant concept of a mixed state. A mixed state \( \mathbf{D} \) de­scribes a situ­ation in which there is not enough in­form­a­tion to de­term­ine the pure state \( \psi \) of the sys­tem. Usu­ally phys­i­cists write \( \mathbf{D} \) as a con­vex com­bin­a­tion of or­tho­gon­al pure states, \( \sum_k w_k \psi_k \) . This nota­tion is con­fus­ing; \( \mathbf{D} \) is not a vec­tor in \( \mathcal{H} \)! It may be in­ter­preted as a list of prob­ab­il­it­ies \( w_k \) that the cor­res­pond­ing pure state is \( \psi_k \). As­so­ci­ated with the state \( \mathbf{D} \) there is a pos­it­ive op­er­at­or \( D \) with trace 1, giv­en by the for­mula \[ D = \sum_k w_k P_k \] where \( P_n \) is the pro­jec­tion on the ei­gen­space of \( \mathbf{D} \) cor­res­pond­ing to the ei­gen­value \( w_n \). The ex­pec­ted value of an ob­serv­able \( A \) is quite clearly \[ E (A) = \sum_k w_k (A\psi_k , \psi_k ) =\operatorname{trace}(D A). \] This is von Neu­mann’s gen­er­al Born rule.

The ei­gen­val­ues of a pro­jec­tion op­er­at­or are 1 and 0; those are the only val­ues a meas­ure­ment of the cor­res­pond­ing ob­serv­able can yield. That is why Mackey calls a pro­jec­tion a ques­tion; the an­swer is al­ways either 1 or 0: “yes” or “no”. The fun­da­ment­al ex­ample is the fol­low­ing. Giv­en a self-ad­joint op­er­at­or \( A \), we will ap­ply the spec­tral the­or­em. Let \( S \) be any Borel sub­set of \( \mathbb{R} \) and let \( P_S \) be the cor­res­pond­ing “spec­tral pro­jec­tion” of \( A \). (If the set \( S \) con­tains only some ei­gen­val­ues of \( A \), then \( P_S \) is simply pro­jec­tion onto the sub­space spanned by the cor­res­pond­ing ei­gen­vectors.) Now sup­pose the state of the sys­tem is the mixed state \( \mathbf{D} \). From the gen­er­al Born rule, the prob­ab­il­ity that a meas­ure­ment of \( A \) lies in \( S \) is the ex­pec­ted value of \( P_S \), namely, \( \operatorname{trace}(DP_S ) \). That is the ob­vi­ous gen­er­al­iz­a­tion of Born’s for­mula for the prob­ab­il­ity that a meas­ure­ment of \( A \) is a par­tic­u­lar ei­gen­value of \( A \) or a set of isol­ated ei­gen­val­ues.

Quite gen­er­ally, con­sider a pos­it­ive op­er­at­or \( D \) with \( \operatorname{trace}(D ) = 1 \). The non­neg­at­ive real-val­ued func­tion \( \mu (P ) = \operatorname{trace}(D P ) \) is a count­ably ad­dit­ive prob­ab­il­ity meas­ure on the lat­tice of pro­jec­tions on \( \mathcal{H} \). This means that if \( \{P_n \} \) is a count­able fam­ily of mu­tu­ally or­tho­gon­al pro­jec­tions, \[ \mu\biggl(\sum_n P_n\biggr) = \sum_n \mu(P_n). \] Also \( \mu (I ) = 1 \). Mackey asked wheth­er every such meas­ure on the pro­jec­tions is of this form, i.e., cor­res­ponds to a state \( D \) . We already men­tioned Mackey’s in­terest in Born’s rule. A pos­it­ive an­swer to Mackey’s ques­tion would show that the Born rule fol­lows from his rather simple ax­ioms for quantum mech­an­ics [e2], [e1], and thus, giv­en these weak pos­tu­lates, Born’s rule is not ad hoc but in­ev­it­able.

Gleason’s theorem

Mackey didn’t try very hard to solve his prob­lem for the ex­cel­lent reas­on that he had no idea how to at­tack it. But he dis­cussed it with a num­ber of ex­perts, in­clud­ing Irving Segal, who men­tioned Mackey’s prob­lem in a gradu­ate class at Chica­go around 1949 or 1950. Among the stu­dents was Dick Kadis­on, who real­ized that there are counter­examples when \( \mathcal{H} \) is two-di­men­sion­al. The high­er-di­men­sion­al case re­mained open.

There mat­ters stood for some years. Then Gleason entered the story. In 1956 he sat in on Mackey’s gradu­ate course on quantum mech­an­ics at Har­vard. To Mackey’s sur­prise, Andy was seized by the prob­lem “with in­tense fe­ro­city”. Moreover, Kadis­on was vis­it­ing MIT at the time, and his in­terest in Mackey’s prob­lem was re­kindled. He quickly per­ceived that there were many “forced inter-re­la­tions” en­tailed by the in­ter­twin­ing of the great circles on the sphere and in prin­ciple a lot could be de­duced from an ana­lys­is of these re­la­tions, though the prob­lem still looked quite tough. He men­tioned his ob­ser­va­tion to Andy, who found it a use­ful hint. (But Kadis­on in­formed me that his ob­ser­va­tion did not in­volve any­thing like Andy’s key “frame func­tion” idea.)

The­or­em 1: (Gleason’s theorem) Let \( \mathcal{H} \) be a sep­ar­able Hil­bert space of di­men­sion great­er than 2. Let \( \mu \) be a count­ably ad­dit­ive prob­ab­il­ity meas­ure on the pro­jec­tions of \( \mathcal{H} \). Then there is a unique non­neg­at­ive self-ad­joint op­er­at­or \( D \), with \( \operatorname{trace}(D ) = 1 \), such that, for every pro­jec­tion \( P \), \[ \mu (P ) = \operatorname{trace}(D P ). \]

The proof has three parts. First, us­ing count­able ad­dit­iv­ity and in­duc­tion, it is easy to re­duce the case of any sep­ar­able real Hil­bert space of di­men­sion great­er than 2 to the 3-di­men­sion­al case. (The com­plex case fol­lows from the real case.)

Next, con­sider a vec­tor \( x \) on the unit sphere. Let \( P_x \) be the one-di­men­sion­al sub­space con­tain­ing \( x \), and define \( f (x) = \mu (P_x ) \). This func­tion is called a frame func­tion. The ad­dit­iv­ity of the meas­ure \( \mu \) im­plies that for any three mu­tu­ally or­tho­gon­al unit vec­tors, \[ f (x) + f (y ) + f (z ) = 1. \]

The proof comes down to show­ing that the frame func­tion \( f \) is quad­rat­ic and there­fore is of the form \( f (x) = \operatorname{trace}(D P_x ) \), where \( D \) is as in the state­ment of the the­or­em. Gleason be­gins his ana­lys­is by show­ing that a con­tinu­ous frame func­tion is quad­rat­ic via a nice piece of har­mon­ic ana­lys­is on the sphere. The center­piece of the pa­per is the proof that \( f \) is con­tinu­ous. Andy told me that this took him most of the sum­mer. It demon­strates his power­ful geo­met­ric in­sight. However, des­pite Andy’s tal­ent for ex­pos­i­tion, much ef­fort is needed to really un­der­stand his ar­gu­ment.

Quite a few people have worked on sim­pli­fy­ing the proof. The pa­per by Cooke, Keane, and Mor­an [e13] is in­ter­est­ing, well writ­ten, and leads the read­er up a gentle slope to Gleason’s the­or­em. The au­thors use an im­port­ant idea of Piron [e9]. (The CKM ar­gu­ment is “ele­ment­ary” be­cause it does not use har­mon­ic ana­lys­is.)

Generalizations of Gleason’s theorem

In his pa­per Andy asked if there were ana­logues of his the­or­em for count­ably ad­dit­ive prob­ab­il­ity meas­ures on the pro­jec­tions of von Neu­mann al­geb­ras oth­er than the al­gebra of bounded op­er­at­ors on sep­ar­able Hil­bert spaces.

A von Neu­mann al­gebra, or \( W^{\ast} \) al­gebra, is an al­gebra \( \mathcal{A} \) of bounded op­er­at­ors on a Hil­bert space \( \mathcal{H} \), closed with re­spect to the ad­joint op­er­a­tion. Most im­port­antly, \( \mathcal{A} \) is closed in the weak op­er­at­or to­po­logy. The lat­ter is defined as fol­lows: a net of bounded op­er­at­ors \( \{a_i \} \) con­verges weakly to \( b \) provided that, for all vec­tors \( x, y \in \mathcal{H} \), \[ \lim_{n\to\infty} (a_nx, y) = (b_x, y). \]

A state of a von Neu­mann al­gebra \( \mathcal{A} \) is a pos­it­ive lin­ear func­tion­al \( \phi : A \to \mathbb{C} \) with \( \phi(I ) = 1 \). This means that \( \phi(x) \geq 0 \) if \( x \geq 0 \) and also \( \|\phi\| = 1 \). The state \( \phi \) is nor­mal provided that if \( a_i \) is an in­creas­ing net of op­er­at­ors that con­verges weakly to \( a \) , then \( \phi(a_i ) \) con­verges to \( \phi(a) \). The nor­mal states on \( B (\mathcal{H}) \) are pre­cisely those of the form \( \operatorname{trace}(D x) \), where \( D \) is a pos­it­ive op­er­at­or with trace 1.

Let \( P (\mathcal{A}) \) be the lat­tice of or­tho­gon­al pro­jec­tions in \( \mathcal{A} \). Then the for­mula \[ \mu (P ) = \phi(P ) \] defines a fi­nitely ad­dit­ive prob­ab­il­ity meas­ure on \( P (\mathcal{A}) \). If \( \phi \) is nor­mal, the meas­ure \( \mu \) is count­ably ad­dit­ive.

The con­verse for count­ably ad­dit­ive meas­ures is due to A. Paszkiewicz [e14]. See E. Christensen [e10] and F. J. Yeadon [e11], [e12] for fi­nitely ad­dit­ive meas­ures. Maeda has a care­ful, thor­ough present­a­tion of the lat­ter in [e16].

It is not sur­pris­ing that the ar­gu­ments use the fi­nite-di­men­sion­al case of Gleason’s the­or­em. A truly easy con­sequence of Gleason’s the­or­em is that \( \mu \) is a uni­formly con­tinu­ous func­tion on the lat­tice of pro­jec­tions \( P \), equipped with the op­er­at­or norm.

A great deal of work has been done on Gleason meas­ures which are un­boun­ded or com­plex val­ued. A good ref­er­ence is [e19]. Bunce and Wright [e18] have stud­ied Gleason meas­ures defined on the lat­tice of pro­jec­tions of a von Neu­mann al­gebra with val­ues in a Banach space. They prove the ana­logue of the res­ults above. A simple ex­ample is Paszkiewicz’s the­or­em for com­plex-val­ued meas­ures, which had been es­tab­lished only for pos­it­ive real-val­ued meas­ures.

Nonseparable Hilbert spaces

Gleason’s the­or­em is true only for sep­ar­able Hil­bert spaces. Robert So­lovay has com­pletely ana­lyzed the non­sep­ar­able case. (Un­pub­lished. However, [e22] is an ex­ten­ded ab­stract.) I con­sider So­lovay’s work to be the most ori­gin­al ex­ten­sion of Gleason’s the­or­em.

Defin­i­tions: A Gleason meas­ure on a Hil­bert space \( \mathcal{H} \) is a count­ably ad­dit­ive prob­ab­il­ity meas­ure on the lat­tice of pro­jec­tions of \( \mathcal{H} \). We say that a Gleason meas­ure \( \mu \) is stand­ard provided there is a pos­it­ive trace-class op­er­at­or \( D \) with trace 1 such that \( \mu (P ) =\operatorname{trace}(D P ) \). Oth­er­wise, \( \mu \) is exot­ic.
Defin­i­tion: A set \( X \) is gi­gant­ic if there is a con­tinu­ous prob­ab­il­ity meas­ure \( \rho \) defined on all the sub­sets of \( X \). Con­tinu­ity means that every point has meas­ure 0.

A count­able set is not gi­gant­ic. In­deed, gi­gant­ic sets are very, very large. Also, in stand­ard set the­or­et­ic ter­min­o­logy, a gi­gant­ic car­din­al is called a meas­ur­able car­din­al.

Gleason’s the­or­em states that every Gleason meas­ure on a sep­ar­able Hil­bert space is stand­ard. But sup­pose \( \mathcal{H} \) is a non­sep­ar­able Hil­bert space with a gi­gant­ic or­thonor­mal basis \( \{e_i : i \in I \} \). Let \( \rho \) be the as­so­ci­ated meas­ure on \( I \). Then the for­mula \[ \mu(P) = \int_I (Pe_i , e_i)\,d\rho(i) \] defines an exot­ic Gleason meas­ure, be­cause \( \mu (Q) = 0 \) for every pro­jec­tion \( Q \) with fi­nite-di­men­sion­al range.

On the oth­er hand, it can be shown that if \( \mathcal{H} \) is any Hil­bert space of nongi­gant­ic di­men­sion great­er than 2, then every Gleason meas­ure on \( \mathcal{H} \) is stand­ard. So­lovay presents a proof. (A con­sequence is that an exot­ic Gleason meas­ure ex­ists if and only if a meas­ur­able car­din­al ex­ists.)

If \( I \) is any set, gi­gant­ic or not, and \( \rho \) is any prob­ab­il­ity meas­ure, con­tinu­ous or not, defined on all the sub­sets of \( I \), then the for­mula above defines a Gleason meas­ure. So­lovay’s main the­or­em says that every Gleason meas­ure is of this form.

The­or­em 2: (Solovay.) Let \( \mathcal{H} \) be a non­sep­ar­able Hil­bert space, and let \( \mu \) be a Gleason meas­ure on \( \mathcal{H} \). Then there is an or­thonor­mal basis \( \{e_i : i \in I \} \) of \( \mathcal{H} \) and a prob­ab­il­ity meas­ure \( \rho \) on the sub­sets of \( I \) such that \( \mu \) is giv­en by the for­mula above.

Ob­serve that Gleason’s the­or­em is ana­log­ous; \( \rho \) is a dis­crete prob­ab­il­ity meas­ure on the in­tegers; the num­bers \( \rho (n ) \) are the ei­gen­val­ues, re­peated ac­cord­ing to mul­ti­pli­city, of the op­er­at­or \( D \).

So­lovay also proves a beau­ti­ful for­mula giv­ing a ca­non­ic­al rep­res­ent­a­tion of a Gleason meas­ure \( \mu \) as an in­teg­ral over the set \( \mathcal{T} \) of pos­it­ive trace-class op­er­at­ors \( A \) of trace 1: there is a meas­ure \( \nu \) defined on all sub­sets of \( \mathcal{T} \) such that, for all \( P \), \[ \mu(P) =\int_{\mathcal{T}}\operatorname{trace}(AP) \,d\nu(A). \] Moreover, there is a unique “pure, sep­ar­ated” meas­ure \( \nu \) such that the for­mula above holds. These two tech­nic­al terms mean that \( \nu \) is sim­il­ar to the sort of meas­ure that oc­curs in spec­tral mul­ti­pli­city the­ory for self-ad­joint op­er­at­ors. The read­er may en­joy prov­ing this for­mula when \( \mathcal{H} \) is fi­nite-di­men­sion­al; this simple case sheds some light on the gen­er­al case.

Hidden variables and the work of John Bell

The ma­jor sci­entif­ic im­pact of Gleason’s the­or­em is not in math­em­at­ics but in phys­ics, where it has played an im­port­ant role in the ana­lys­is of the basis of quantum mech­an­ics. A ma­jor ques­tion is wheth­er prob­ab­il­ist­ic quantum mech­an­ics can be un­der­stood as a phe­nomen­o­lo­gic­al the­ory ob­tained by av­er­aging over vari­ables from a deep­er non­prob­ab­il­ist­ic the­ory. The the­ory of heat ex­em­pli­fies what is wanted. Heat is now un­der­stood as due to the col­li­sions of atoms and mo­lecules. In this way one can un­der­stand ther­mo­dy­nam­ics as a phe­nomen­o­lo­gic­al the­ory de­rived by av­er­ages over “hid­den vari­ables” as­so­ci­ated with the deep­er particle the­ory; hence the term “stat­ist­ic­al mech­an­ics”. Ein­stein sought an ana­log­ous re­la­tion between quantum mech­an­ics and — what? He is sup­posed to have said that he had giv­en one hun­dred times more thought to quantum the­ory than to re­lativ­ity.

The fourth chapter of John von Neu­mann’s great book [e7] is de­voted to his fam­ous ana­lys­is of the hid­den vari­able ques­tion. His con­clu­sion was that no such the­ory could ex­ist. He writes, “The present sys­tem of quantum mech­an­ics would have to be ob­ject­ively false, in or­der that an­oth­er de­scrip­tion of the ele­ment­ary pro­cesses than the stat­ist­ic­al one may be pos­sible.” That seemed to settle the ques­tion. Most phys­i­cists wer­en’t much in­ter­ested in the first place when ex­cit­ing new dis­cov­er­ies were al­most shower­ing down.

But in 1952 there was a sur­prise. Con­trary to von Neu­mann, Dav­id Bo­hm ex­hib­ited a hid­den vari­able the­ory by con­struct­ing a sys­tem of equa­tions with both waves and particles which ex­actly re­pro­duced quantum mech­an­ics. But Ein­stein re­jec­ted this the­ory as “too easy”, be­cause it lacked the in­sight Ein­stein was seek­ing. Worse yet, it had the fea­ture Ein­stein most dis­liked. Ein­stein had no prob­lem un­der­stand­ing that there can eas­ily be cor­rel­a­tions between the be­ha­vi­or of two dis­tant sys­tems, \( A \) and \( B \). If there is a cor­rel­a­tion due to in­ter­ac­tion when the sys­tems are close, it can cer­tainly be main­tained when they fly apart. His ob­jec­tion to stand­ard quantum mech­an­ics was that in some cases a meas­ure­ment of sys­tem \( A \) in­stantly de­term­ines the res­ult of a re­lated meas­ure­ment of sys­tem \( B \). Ein­stein dubbed this “weird ac­tion at a dis­tance.” Bo­hm’s mod­el has this ob­jec­tion­able prop­erty.

In fact, soon after its pub­lic­a­tion, von Neu­mann’s ar­gu­ment was de­mol­ished by Grete Her­mann [e8], a young stu­dent of Emmy No­eth­er. Her point was that in quantum mech­an­ics the ex­pect­a­tion of the sum of two ob­serv­ables \( A \) and \( B \) is the sum of the ex­pect­a­tions: \( E (A + B ) = E (A) + E (B ) \), even if \( A \) and \( B \) don’t com­mute. This is a “mir­acle” be­cause the ei­gen­val­ues of \( A + B \) have no re­la­tion to those of \( A \) and \( B \) un­less \( A \) and \( B \) com­mute. It is true only be­cause of the spe­cial for­mula for ex­pect­a­tions in quantum mech­an­ics. It is not a “law of thought”. Yet von Neu­mann pos­tu­lated that ad­dit­iv­ity of ex­pec­ted val­ues must hold for all un­der­ly­ing hid­den vari­able the­or­ies. That is the fatal mis­take in von Neu­mann’s ar­gu­ment. However, al­though Heis­en­berg im­me­di­ately un­der­stood Her­mann’s ar­gu­ment when she spoke with him, her work was pub­lished in an ob­scure journ­al and was for­got­ten for dec­ades.

The out­stand­ing Ir­ish phys­i­cist John Bell was ex­tremely in­ter­ested in the hid­den vari­able prob­lem. Early on he dis­covered a simple ex­ample of a hid­den vari­able the­ory for a two-di­men­sion­al quantum sys­tem; it’s in chapter 1 of [e15], which is a re­print of [e4]. This is an­oth­er counter­example for von Neu­mann’s “im­possib­il­ity” the­or­em. (Bell did a great deal of im­port­ant “re­spect­able” phys­ics. He said that he stud­ied the philo­sophy of phys­ics only on Sat­urdays. An in­ter­est­ing es­say on Bell is in Bern­stein’s book [e17].)

When Bell learned of Gleason’s the­or­em he per­ceived that in Hil­bert spaces of di­men­sion great­er than 2, it “ap­par­ently” es­tab­lishes von Neu­mann’s “no hid­den vari­ables” res­ult without the ob­jec­tion­able as­sump­tions about non­com­mut­ing op­er­at­ors. Bell is re­por­ted to have said that he must either find an “in­tel­li­gible” proof of Gleason’s the­or­em or else quit the field. For­tu­nately Bell did de­vise a straight­for­ward proof of a very spe­cial case: nonex­ist­ence of frame func­tions tak­ing only the val­ues 0 and 1. Such frame func­tions cor­res­pond to pro­jec­tions. This case suf­ficed for Bell’s pur­poses [e4]. See the first chapter of [e15].2

At the International Conference on the Teaching of Mathematics, Samos (home of Pythagoras), 1998.
Photo courtesy of Jean Berko Gleason.

The gist of von Neu­mann’s proof is an ar­gu­ment that dis­per­sion-free states do not ex­ist. Here a state \( D \) is dis­per­sion-free provided \( E (A^2 ) = E (A)^2 \) for any ob­serv­able \( A \). In oth­er words, every ob­ser­va­tion of \( A \) has the value \( E (A) \), its mean value. Quantum mech­an­ics is sup­posedly ob­tained by av­er­aging over such states. The frame func­tions con­sidered by Bell cor­res­pond pre­cisely to dis­per­sion-free states. But these frame func­tions are not con­tinu­ous. Gleason’s the­or­em im­plies that no such frame func­tions ex­ist. There­fore there are no dis­per­sion-free states. But Gleason’s the­or­em uses Mackey’s pos­tu­late of ad­dit­iv­ity of ex­pect­a­tions for com­mut­ing pro­jec­tions. Bell’s ar­gu­ment based on Gleason’s the­or­em avoids the un­jus­ti­fied as­sump­tion of ad­dit­iv­ity of ex­pect­a­tion val­ues for non­com­mut­ing op­er­at­ors.

Bell writes: “That so much fol­lows from such ap­par­ently in­no­cent as­sump­tions leads one to ques­tion their in­no­cence.” He points out that if \( P \), \( Q \), and \( R \) are pro­jec­tions with \( P \) and \( Q \) or­tho­gon­al to \( R \) but not to each oth­er, we might be able to meas­ure \( R \) and \( P \), or \( R \) and \( Q \), but not ne­ces­sar­ily both, be­cause \( P \) and \( Q \) do not com­mute. Con­cretely, the two sets of meas­ure­ments may well re­quire dif­fer­ent ex­per­i­ment­al ar­range­ments. (This point was of­ten made by Niels Bo­hr.) Bell ex­presses this fun­da­ment­al fact em­phat­ic­ally: “The danger in fact was not in the ex­pli­cit but in the im­pli­cit as­sump­tions. It was ta­citly as­sumed that meas­ure­ment of an ob­serv­able must yield the same value in­de­pend­ently of what oth­er meas­ure­ments are made sim­ul­tan­eously.” In oth­er words, the meas­ure­ment may de­pend on its con­text. This amounts to say­ing that Gleason’s frame func­tions may not be well defined from the point of view of ac­tu­al ex­per­i­ments. Ac­cord­ingly, one should ex­am­ine Mackey’s ap­par­ently plaus­ible de­riv­a­tion that pro­jec­tion-val­ued meas­ures truly provide part of a val­id ax­io­mat­iz­a­tion of quantum mech­an­ics.

Fi­nally, a few words about the fam­ous “Bell’s In­equal­ity”.

The second chapter of Bell’s book is a re­print of [e3] (ac­tu­ally writ­ten after [e4]). In this very im­port­ant pa­per, Bell de­rives a spe­cif­ic in­equal­ity sat­is­fied by cer­tain “loc­al” hid­den vari­able the­ory for non­re­lativ­ist­ic quantum mech­an­ics. (“Loc­al­ity” ex­cludes “weird” cor­rel­a­tions of meas­ure­ments of widely sep­ar­ated sys­tems.) There are many sim­il­ar but more gen­er­al in­equal­it­ies. Moreover, the study of the “en­tan­gle­ment” of sep­ar­ated quantum sys­tems has opened a new field of math­em­at­ic­al re­search.

Gleason in Egypt in 2001.
Photo courtesy of Jean Berko Gleason.

Start­ing in 1969, dif­fi­cult ex­per­i­ment­al work began, us­ing vari­ants of Bell’s in­equal­ity, to test if very del­ic­ate pre­dic­tions of quantum mech­an­ics are cor­rect. Of course, quantum mech­an­ics has giv­en su­perb ex­plan­a­tions of all sorts of phe­nom­ena, but these ex­per­i­ments wa­ter­board quantum mech­an­ics. Many ex­per­i­ments have been done; so far there is no con­vin­cing evid­ence that quantum mech­an­ics is in­cor­rect. In ad­di­tion, ex­per­i­ments have been done which sug­gest that in­flu­ence from one sys­tem to the oth­er propag­ates enorm­ously faster than light. These ex­per­i­ments point to­ward in­stant­an­eous trans­fer of in­form­a­tion.

Bell’s pa­pers on quantum philo­sophy have been col­lec­ted in his book Speak­able and Un­speak­able in Quantum Mech­an­ics [e15]. The first pa­per [e4] dis­cusses Gleason’s the­or­em and the second “Bell’s in­equal­ity”. The en­tire book is a pleas­ure to read.3


Among his many tal­ents, Andy was a mas­ter of ana­grams. His frag­ment­ary 1947 di­ary re­cords a fam­ily vis­it dur­ing Har­vard’s spring break:

March 30.  …We played ana­grams after sup­per and I won largely through the char­ity of the op­pos­i­tion.

April 1.  …Played a game of ana­grams with Moth­er and won.

April 2.  …Moth­er beat me to­night at ana­grams.

So we know a little about where he honed that tal­ent.

Many years ago Andy and I had a little ana­gram “con­test” by mail. (Dick Kadis­on said then, “You’re hav­ing an ana­gram com­pet­i­tion with Andy Gleason? That’s like arm wrest­ling with Gar­gan­tua.”) Any­how, I figured out ROAST MULES, and I was proud to come up with I AM A WON­DER AT TANGLES, which is an ana­gram of AN­DREW MAT­TAI GLEASON. Un­for­tu­nately, it should be MAT­TEI. But I didn’t have the chutzpah to ask Andy to change the spelling of his middle name. I am grate­ful for very in­ter­est­ing cor­res­pond­ence and con­ver­sa­tions with the late Andy Gleason and George Mackey, to­geth­er with Dick Kadis­on, Si Kochen, and Bob So­lovay.


[1]A. M. Gleason: “Meas­ures on the closed sub­spaces of a Hil­bert space,” J. Math. Mech. 6 : 4 (1957), pp. 885–​893. MR 0096113 Zbl 0078.​28803 article

[2]An­drew M. Gleason: Glimpses of a life in math­em­at­ics. Edi­ted by E. Bolk­er, P. Chernoff, C. Costes, and D. Lieber­man. Privately prin­ted, 1992. book