Annals of Mathematics
The Logic of Quantum Mechanics
Author(s): Garrett Birkhoff and John Von Neumann
Source: Annals of Mathematics, Second Series, Vol. 37, No. 4 (Oct., 1936), pp. 823-843
Published by: Annals of Mathematics
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ANNALS
OF MATHEMATICS
Vol. 37, No. 4, October,1936
THE LOGIC OF QUANTUM MECHANICS
By GARRETT BIRKHOFF AND JOHN VON NEUMANN
(Received April4, 1936)
1. Introduction. One of the aspects of quantumtheorywhichhas attracted
the most generalattention,is the noveltyof the logical notions which it presupposes. It assertsthat even a completemathematicaldescriptionof a physical system' does not in generalenable one to predictwithcertaintythe result
of an experimenton I, and that in particularone can never predictwith certaintyboth the positionand the momentumof (E (Heisenberg'sUncertainty
Principle). It furtherassertsthat mostpairs of observationsare incompatible,
and cannot be made on ' simultaneously(Principleof Non-commutativity
of
Observations).
The object of the presentpaper is to discoverwhat logicalstructureone may
hope to findin physicaltheorieswhich,like quantum mechanics,do not conformto classical logic. Our main conclusion,based on admittedlyheuristic
arguments,is that one can reasonablyexpect to finda calculus of propositions
whichis formallyindistinguishable
fromthe calculus of linear subspaces with
respectto setproducts,linearsums,and orthogonal
complements-andresembles
the usual calculusof propositionswithrespectto and, or,and not.
In orderto avoid beingcommittedto quantumtheoryin its presentform,we
have first(in ??2-6) stated the heuristicargumentswhichsuggestthat such a
calculus is the properone in quantum mechanics,and then (in ??7-14) reconstructedthiscalculusfromthe axiomaticstandpoint. In bothpartsan attempt
has been made to clarifythe discussionby continualcomparisonwithclassical
mechanicsand its propositionalcalculi. The paper ends with a fewtentative
conclusionswhichmay be drawnfromthe materialjust summarized.
I. PHYSICAL BACKGROUND
2. Observationson physicalsystems. The conceptof a physicallyobserv-
able "physical system" is present in all branches of physics, and we shall
assume it.
It is clear that an "observation" of a physicalsystem( can be described
generallyas a writingdown of the readingsfromvarious' compatiblemeasurements. Thus ifthe measurements
are denotedby the symbolsAl, *, An) then
.
1 If one prefers,
one mayregarda set ofcompatiblemeasurements
as a singlecomposite
"measurement"-andalso admitnon-numerical
withsubsereadings-withoutinterfering
quentarguments.
Amongconspicuousobservablesin quantumtheoryare position,momentum,
energy,
and (non-numerical)
symmetry.
823
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824
GARRETT
BIRKHOFF
AND JOHN VON NEUMANN
an observationof S amountsto specifyingnumbersxi, *. *,x, corresponding
to the different
ilkIt followsthat the mostgeneralformof a predictionconcerning( is that the
point(xi, ***, x,) determined
by actuallymeasuring
Al, ** , ;z, will lie in a
subset S of (xi, ***, xn)-space. Hence ifwe call the (xi, **, xn)-spacesassociatedwith(, its "observation-spaces,"we may call the subsetsof the observation-spacesassociated with any physicalsystemA, the "experimentalpropositions"concerning 5.
.
3. Phase-spaces. There is one conceptwhichquantum theoryshares alike
withclassicalmechanicsand classicalelectrodynamics. This is the conceptofa
mathematical"phase-space."
Accordingto this concept,any physicalsystem25 is at each instantlyhypotheticallyassociated with a "point" p in a fixedphase-space 2; this point is
supposedto representmathematicallythe "state" of 5,and the "state" of e is
supposedto be ascertainableby "maximal"2observations.
Furthermore,
the pointpoassociatedwithe at a timeto,togetherwitha prescribedmathematical"law of propagation,"fixthe point pi associated with e
at any later time t; this assumptionevidentlyembodiesthe principleof mathematicalcausation.3
Thus in classical mechanics,each point of I. correspondsto a choice of n
positionand n conjugatemomentumcoordinates-and the law of propagation
may be Newton's inverse-squarelaw of attraction. Hence in this case 2 is a
iegion of ordinary2n-dimensional
space. In electrodynamics,
the points of 2
can only be specifiedaftercertainfunction8-suchas the electromagnetic
and
electrostatic
potential-are known;hence2 is a function-space
ofinfinitely
many
dimensions. Similarly,in quantumtheorythepointsofZ correspondto so-called
"wave-functions,"
and hence Z is again a function-space-usually4
assumed to
be Hilbertspace.
In electrodynamics,
the law of propagationis containedin Maxwell's equations,and in quantum theory,in equations due to Schrodinger. In any case,
the law of propagationmay be imaginedas inducinga steady fluidmotionin
the phase-space.
It has proved to be a fruitfulobservationthat in many importantcases of
classical dynamics,this flow conservesvolumes. It may be noted that in
quantummechanics,the flowconservesdistances(i.e., the equations are "unitary").
2 L. Pauling and E. B. Wilson, "An introductionto quantum mechanics," McGraw-Hill,
1935,p. 422. Dirac, "Quantummechanics,"Oxford,1930,?4.
3For theexistenceofmathematical
causation,cf.also p. 65 ofHeisenberg's"The physical
principles of the quantum theory,"Chicago, 1929.
" Cf. J. von Neumann, "MathematischeGrundlagender Quanten-mechanik,"Berlin, 1931.
p. 18.
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LOGIC OE QUANTUM MECHANICS
825
4. Propositions
as subsetsof phase-space. Now beforea phase-spacecan
become imbued with reality,its elementsand subsets must be correlatedin
some way with "experimentalpropositions" (which are subsets of different
observation-spaces). Moreover,thismustbe so done that set-theoretical
inclusion (whichis theanalogueoflogicalimplication)is preserved.
There is an obviousway to do thisin dynamicalsystemsofthe classicaltype.'
One can measure position and its firsttime-derivativevelocity-and hence
momentum-explicitly,and so establisha one-one correspondencewhich preservesinclusionbetweensubsetsof phase-spaceand subsetsof a suitable observation-space.
In the cases of the kinetictheoryof gases and of electromagnetic
waves no
such simple procedureis possible, but it was imagined for a long time that
"demons" of small enoughsize could by tracingthe motionof each particle,or
by a dynamometer
and infinitesimal
point-charges
and magnets,measurequantitiescorresponding
to everycoordinateofthe phase-spaceinvolved.
In quantumtheorynot even thisis imagined,and the possibilityofpredicting
in generalthereadingsfrommeasurements
on a physicalsysteme froma knowledge ofits "state" is denied;onlystatisticalpredictionsare alwayspossible.
This has been interpretedas a renunciationof the doctrineof pre-determination; a thoughtful
analysisshowsthat anotherand moresubtleidea is involved.
The centralidea is thatphysicalquantitiesare related,
but are not all computable
froma numberofindependent
basicquantities(such as positionand velocity).'
We shall showin ?12 that thissituationhas an exact algebraicanalogue in the
calculusofpropositions.
5. Propositionalcalculi in classical dynamics. Thus we see that an uncritical acceptance of the ideas of classical dynamics (particularlyas they
involven-bodyproblems)leads one to identifyeach subset of phase-spacewith
an experimentalproposition(the propositionthat the systemconsideredhas
position and momentumcoordinatessatisfyingcertain conditions) and conversely.
This is easily seen to be unrealistic;forexample,how absurd it would be to
call an "experimentalproposition,"the assertionthat the angular momentum
(in radiansper second) of the eartharoundthe sun was at a particularinstanta
rationalnumber!
Actually,at least in statistics,it seemsbest to assume that it is the Lebesguemeasurablesubsetsof a phase-spacewhichcorrespondto experimentalproposi0.7
iftheirdifference
has Lebesgue-measure
tions,twosubsetsbeingidentified,
' Like systemsidealizingthesolarsystemorprojectilemotion.
' A similarsituationariseswhenone triesto correlatepolarizationsin different
planesof
electromagnetic
waves.
7 Cf. J. von Neumann,"Operatorenmethodenin der klassischen Mechanik," Annals of
Math. 33 (1932),595-8. The difference
of two sets Si, St is the set (St + S2) - St-2 of
thosepoints,whichbelongto one ofthem,but notto both.
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826
GARRETT
BIRKHOFF
AND JOHN VON NEUMANN
But in eithercase, theset-theoretical
sumand productofany two subsets,and
to experimental
the complementofany one subsetofphase-spacecorresponding
propositions,
has the same property. That is, by definition8
The experimental
propositionsconcerningany systemin classical mechanics,
correspondto a "field" of subsetsof its phase-space. More precisely:To the
"quotient"of such a fieldby an ideal in it. At any rate theyforma "Boolean
Algebra."9
In the axiomaticdiscussionof propositionalcalculi whichfollows,it will be
shownthat this is inevitablewhen one is dealing with exclusivelycompatible
measurements,
and also that it is logicallyimmaterialwhichparticularfieldof
setsis used.
6. A propositionalcalculus for quantum mechanics. The question of the
connectionin quantum mechanicsbetweensubsets of observation-spaces(or
"experimentalpropositions")and subsets of the phase-spaceof a systema,
has not been touched. The presentsectionwill be devoted to definingsuch a
connection,provingsome factsabout it, and obtainingfromit heuristicallyby
introducinga plausible postulate, a propositionalcalculus for quantum mechanics.
Accordingly,
let us observethatifal,
, an are any compatibleobservations
on a quantum-mechanical
systemS withphase-spaceX, then'0thereexistsa set
ofmutuallyorthogonalclosedlinearsubspaces Uj of Z (whichcorrespondto the
familiesof properfunctionssatisfyingaif = Xi.if,-... anf = Xi.,f) such that
every
point(orfunction)f e Z can be uniquelywrittenin theform
.
celf +
f =
C2f2 +
cafS +
*[fi
e(
Hence ifwe state the
DEFINITION:
By the "mathematicalrepresentative"of a subset S of any
observation-space(determinedby compatibleobservationsai, - - , a") for a
quantum-mechanical
system 5,willbe meanttheset ofall pointsf ofthephasespace of (5, which are linearlydeterminedby properfunctionsfk satisfying
alfk =
Xlfk,
* * -*
afk
=
Xnfk,
where(X1,***,
Xn) e S.
Then it followsimmediately:(1) that the mathematicalrepresentativeof any
experimentalpropositionis a closed linear subspace of Hilbert space (2) since
all operatorsof quantum mechanics are Hermitian,that the mathematical
representative
of the negative"of any experimentalpropositionis the orthogonal
F. Hausdorff,
"Mengenlehre,"
Berlin,1927,p. 78.
H. Stone,"BooleanAlgebrasand theirapplicationtotopology,"
Proc. Nat. Acad. 20
(1934),p. 197.
10Cf.von Neumann,op. cit., pp. 121,90,or Dirac, op. cit., 17. We disregardcomplications due to the possibilityof a continuousspectrum. They are inessentialin the present case.
11By the "negative" of an experimentalproposition(or subset S of an observationofS in
to the sef-complement
space) is meantthe experimental
propositioncorresponding
the same observation-space.
8
9M.
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LOGIC OF QUANTUMMECHANICS
827
of the mathematicalrepresentativeof the propositionitself(3) the
complement
threeconditionson two experimentalpropositionsP and Q concerning
following
a giventypeofphysicalsystemare equivalent:
(3a) The mathematicalrepresentativeof P is a subset of the mathematical
ofQ.
representative
(3b) P impliesQ-that is, wheneverone can predictP withcertainty,one can
predictQ with certainty.
(3c) For any statisticalensembleof systems,the probabilityof P is at most
theprobabilityofQ.
The equivalenceof (3a)-(3c) leads one to regardthe aggregateof the mathepropositionsconcerningany physical
oftheexperimental
maticalrepresentatives
thepropositionalcalculusforS.
mathematically
system(B, as representing
We nowintroducethe
representatives
productof any twomathematical
POSTULATE: The set-theoretical
is itselfthe
system,
a quantum-mechanical
concerning
propositions
ofexperimental
proposition.
ofan experimental
representative
mathematical
REMARKS: This postulate would clearly be implied by the not unnatural
operatorsin Hilbert space (phaseconjecturethat all Hermitian-symmetric
space) correspondto observables;12 it would even be impliedby the conjecture
that thoseoperatorswhichcorrespondto observablescoincidewiththe HermiM.13
elementsof a suitableoperator-ring
tian-symmetric
Now the closed linearsum Q%+ Q2of any two closed linear subspaces Q. of
Hilbert space, is the orthogonalcomplementof the set-producta'- 2' of the
henceifone adds theabove postulateto the
Q. ofthe Qua;
orthogonalcomplements
then
one can deducethat
of
theory,
usual postulates quantum
complement
and closedlinearsumofany two,and theorthogonal
The set-product
an
mathematically
of any one closedlinearsubspaceof Hilbertspace representing
systemS, itself
experimentalpropoition concerninga quantum-mechanical
concerning
(S.
proposition
an experimental
represents
This defines the calculus of experimental propositions concerning A5, as a
calculus with three operations and a relation of implication,which closely
resemblesthesystemsdefinedin ?5. We shallnowturnto theanalysisand comstandpoint.
parisonofall threecalculifroman axiomatic-algebraic
II. ALGEBRAIC ANALYSIS
7. Implicationas partial ordering. It was suggested above that in any
physicaltheoryinvolvinga phase-space,the experimentalpropositionsconcern12I.e., that given such an operatora, one "could" findan observableforwhichthe
properstateswerethe properfunctionsofa.
1I F. J. Murrayand J. v. Neumann,"On rings of operators," AnnalsofMath., 37 (1936),
p. 120. It is shownon p. 141,loc. cit. (Definition4.2.1 and Lemma4.2.1), that the closed
linearsets of a ringM-that is those,the "projectionoperators"of whichbelongto Mcoincidewiththeclosedlinearsetswhichare invariantundera certaingroupofrotationsof
Hilbert space. And the latter propertyis obviouslyconservedwhen a set-theoretical
is formed.
intersection
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828
GARRETT
BIRKHOFF
AND JOHN VON NEUMANN
ing a systeme correspondto a familyof subsetsof its phase-space2, in such a
way that "x implies y" (x and y being any two experimentalpropositions)
means that the subset of 2 corresponding
to x is containedset-theoretically
in
the subsetcorresponding
to y. This hypothesisclearlyis importantin proportion as relationshipsof implicationexist between experimentalpropositions
corresponding
to subsetsofdifferent
observation-spaces.
The presentsectionwillbe devotedto corroborating
thishypothesisby identifyingthe algebraic-axiomatic
propertiesof logicalimplicationwiththoseof setinclusion.
It is customaryto admitas relationsof "implication,"onlyrelationssatisfy-
ing
Sl: x impliesx.
S2: If x impliesy and y impliesz, thenx impliesz.
S3: If x impliesy and y impliesx, thenx and y are logicallyequivalent.
In fact,S3 need not be stated as a postulateat all, but can be regardedas a
definition
oflogicalequivalence. Pursuingthislineofthought,one can interpret
as a "physicalquality,"theset ofall experimental
propositionslogicallyequivalentto a givenexperimental
proposition.'4
Now ifone regardstheset S. ofpropositions
implyinga givenpropositionx as
a "mathematicalrepresentative"of x, thenby S3 the correspondence
between
the x and the S. is one-one,and x impliesy if and onlyif S. C S,. While conversely,if L is any systemof subsetsX of a fixedclass r, thenthereis an isomorphismwhichcarriesinclusioninto logical implicationbetweenL and the
systemL* ofpropositions
"x is a pointofX," X e L.
Thus we see that the propertiesof logical implicationare indistinguishable
fromthoseofset-inclusion,
and thattherefore
it is algebraically
reasonableto try
to correlatephysicalqualitieswithsubsetsofphase-space.
A systemsatisfying
S1-S3, and in whichthe relation"x impliesy" is written
x C y,is usually'6called a "partiallyorderedsystem,"and thusour firstpostulate concerningpropositionalcalculi is that thephysicalqualitiesattributable
to
anyphysicalsystem
forma partiallyordered
system.
It doesnotseemexcessiveto requirethatin additionany suchcalculuscontain
two special propositions:the proposition0 that the systemconsideredexists,
and theproposition? thatit does notexist. Clearly
S4:
?
C x C 0 forany x.
? is,froma logicalstandpoint,the"identicallyfalse"or "absurd" proposition;
LIis the "identicallytrue" or "self-evident"proposition.
8. Lattices. In any calculus of propositions,it is natural to imagine that
thereis a weakestpropositionimplying,and a strongestpropositionimpliedby,
14Thus in ?6, closed linearsubspacesof Hilbertspace correspond
one-manyto experimentalpropositions,
but one-oneto physicalqualitiesin thissense.
15 F. Hausdorff,
"Grundzage derMengenlehre,"Leipzig, 1914,Chap. VI, 11.
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LOGIC
OF QUANTUM
MECHANICS
829
a givenpair ofpropositions. In fact,investigationsofpartiallyorderedsystems
fromdifferent
anglesall indicatethat the firstpropertywhichtheyare likelyto
possess,is the existenceofgreatestlowerboundsand least upperboundsto subwe state
sets of theirelements. Accordingly,
DEFINITION:
A partiallyorderedsystemL will be called a "lattice" if and
onlyifto any pair x and y ofits elementstherecorrespond
S5: A "meet" or "greatestlowerbound" x n y such that (5a) x
x ny Cy, (5c)z Cx andz Cy implyzcx n y.
n y C x, (5b)
S6: A "join" or "least upperbound"x nflysatisfying
(6a) x U y D x, (6b)
x U y D y, (6c) w D x andw D y implyw D x U y.
The relationbetween meets and joins and abstract inclusioncan be summarized as follows,'6
(8.1) In any latticeL, thefollowingformalidentitiesare true,
n a = a and a U a = a.
n b = b n a and a U b = b U a.
afn (b n c) = (afn b) n c and a U (b U c) = (a U b) U c.
a U (a n b) = a n (a U b) = a.
Moreover,the relationsa D b, a n b = b, and a U b = a are equivalent-each
Li:
L2:
L3:
L4:
a
a
impliesbothofthe others.
(8.2) Conversely,in any set of elementssatisfyingL2-L4 (L1 is redundant),
a n b = b and a U b = a are equivalent. And if one definesthemto mean
a D b,thenone revealsL as a lattice.
Clearly L1-L4 are well-knownformalpropertiesof and and or in ordinary
logic. This gives an algebraicreasonforadmittingas a postulate(ifnecessary)
the statementthat a givencalculusofpropositionsis a lattice. There are other
reasons17
whichimpelone to admitas a postulatethestrongerstatementthat the
set-productof any two subsets of a phase-spacewhich correspondto physical
qualities, itselfrepresentsa physicalquality-this is, of course,the Postulate
of ?6.
It is worthremarkingthat in classical mechanics,one can easily definethe
meet or join of any two experimentalpropositionsas an experimental
proposition-simplyby havingindependentobserversread offthe measurementswhich
eitherpropositioninvolves,and combiningthe resultslogically. This is truein
quantum mechanicsonly exceptionally-onlywhen all the measurementsinvolved commute(are compatible);in general,one can only expressthe join or
16The finalresultwas foundindependently
by 0. Ore, "The foundationsof abstract
algebra. I.," AnnalsofMath. 36 (1935),406-37,and by H. MacNeille in his Harvard DoctoralThesis, 1935.
17
The firstreason is that this implies no restrictionon the abstract nature of a lattice-
anylatticecan be realizedas a systemofits ownsubsets,in sucha waythata n b is thesetproductofa and b. The secondreasonis thatifoneregardsa subsetS ofthephase-spaceof
ofobserving(S in S, thenit is naturalto assume
to thecertainty
a systeme as corresponding
that the combinedcertaintyof observingS in S and T is the certaintyof observing ; in
So T = SfnT,-and assumesquantumtheory.
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830
GARRETT
BIRKHOFF
AND JOHN VON NEUMANN
meet of two given experimentalpropositionsas a class of logicallyequivalent
experimentalpropositions-i.e.,as a physicalquality."8
9. Complementedlattices. Besides the (binary) operationsof meet- and
join-formation,
thereis a third(unary) operationwhichmay be definedin partiallyorderedsystems. This is theoperationofcomplementation.
In thecase oflatticesisomorphicwith"fields"ofsets,complementation
corresponds to passage to the set-complement.In the case of closed linear subspaces ofHilbertspace (or of Cartesiann-space),it correspondsto passage to the
orthogonalcomplement. In either case, denotingthe "complement" of an
elementa by a', one has theformalidentities,
L71: (a')' = a.
L72: a n a' = ? and a U a'
L73: a C b impliesa' D b'.
=
:
By definition,
L71 and L73 amount to assertingthat complementationis a
ofperiodtwo. It is an immediatecorollaryofthisand the
"dual automorphism"
dualitybetweenthe definitions
(in termsofinclusion)ofmeetand join, that
L74: (a n b)' = a' U b' and (a U b)' = a', b'
and anothercorollarythat the secondhalfofL72 is redundant. [Proof:by L71
and the firsthalfof L74, (a U a') = (a" U a') = (a' n a)' = ?', whileunder
inversionof inclusion? evidentlybecomes0.] This permitsone to deduceL72
fromtheevenweakerassumptionthat a C a' impliesa - . Proof:forany x,
(x fnxI)I = (' u x") = x' u x Dxf n x'.
Hence if one admitsas a postulatethe assertionthat passagefroman experimentalproposition
a toitscomplement
a' is a dual automorphism
ofperiodtwo,and
a impliesa' is absurd,one has in effectadmittedL71-L74.
This postulateis independentlysuggested(and L71 proved) by the fact the
"complement"of the propositionthat the readingsxi, ***, x, froma seriesof
compatibleobservationsyA1,** , /Anliein a subsetS of (xi, **- , xn)-space,is by
thepropositionthatthereadingslie in theset-complement
ofS.
definition
10. The distributiveidentity. Up to now, we have only discussed formal
featuresoflogicalstructurewhichseemto be commonto classicaldynamicsand
thequantumtheory. We nowturnto the centraldifference
betweenthem-the
distributive
identity
ofthepropositionalcalculus:
L6: a U (b n c)
=
(a U b)
n (a U c) anda
fn (b U c) = (a Rb) U (afn
c)
whichis a law in classical,but notin quantummechanics.
18 The following
If a, b
pointshouldbe mentionedin orderto avoid misunderstanding:
are twophysicalqualities,thena U b,an b and a' (cf.below) are physicalqualitiestoo (and
so are 0 and 0+). But a C b is not a physicalquality; it is a relationbetweenphysical qualities.
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LOGIC
831
MECHANICS
OF QUANTUM
From an axiomaticviewpoint,each half of L6 impliesthe other.'9 Further,
eitherhalfof L6, taken withL72, impliesL71 and L73, and to assume L6 and
of a Boolean algebra.20
L72 amountsto assumingtheusual definition
From a deeper mathematicalviewpoint,L6 is the characteristicpropertyof
set-combination. More precisely,every "field" of sets is isomorphicwith a
Boolean algebra, and conversely.21 This throwsnew lighton the well-known
fact that the propositionalcalculi of classical mechanicsare Boolean algebras.
ofthe
It is interesting
thatL6 is also a logicalconsequenceofthecompatibility
observablesoccurringin a, b, and c. That is, if observationsare made by independentobservers,and combinedaccordingto the usual rules of logic,one can
proveLi-L4, L6, and L71-74.
These facts suggestthat the distributivelaw may break down in quantum
mechanics. That it doesbreak down is shownby the factthat if a denotesthe
experimentalobservationof a wave-packet4I'on one side of a plane in ordinary
on the otherside,and b the obserthe observationof4fr
space, a' correspondingly
about theplane,then(as one can readilycheck):
vationof6 in a statesymmetric
f
b n (a U a') = bn
= b>
?
= (b
n a)
= (bna)
= (b
n a')
U (bna')
REMARK:
In connectionwith this, it is a salient fact that the generalized
law oflogic:
distributive
in
L6*: II
n
i
\m
E aid)
j
1
=
E
jgi)
(f[
i
1
\
aij(i))
breaksdownin the quotientalgebraof the fieldof Lebesgue measurablesets by
theideal ofsets ofLebesguemeasure0, whichis so fundamentalin statisticsand
oftheergodicprinciple.s
theformulation
11. The modular identity. Although closed linear subspaces of Hilbert
space and Cartesian n-space need not satisfyL6 relativeto set-productsand
closedlinearsums,the formalpropertiesof theseoperationsare not confinedto
L1-L4 and L71-L73.
to satisfythe
In particular,set-productsand straightlinearsumsare known22
so-called"modularidentity."
19
1931,vol. 2, p. 110.
R. Dedekind,"Werke,"Braunschweig,
" G. Birkhoff,"On the combination of subalgebras," Proc. Camb. Phil. Soc. 29 (1933),
withrespectto inclusion
441-64,??23-4. Also, in any lattice satisfyingL6, isomorphism
this need not be trueif L6 is not
impliesisomorphism
withrespectto complementation;
assumed,as thlelatticeoflinearsubspacesthroughthe originof Cartesiann-spaceshows.
21
M. H. Stone, "Boolean algebras and theirapplication to topology," Proc. Nat. Acad. 20
(1934),197-202.
22 A detailedexplanationwill be omitted,forbrevity;one could refer
to workof G. D.
Birkhoff,
J. von Neumann,and A. Tarski.
23 G. Birkhoff,
op. cit., ?28. The proofis easy. One firstnotesthat sincea C (a U b) nc
ifaCc,andbncC(aUb)
(bnfc) C (aU b) nc. Thenonenotes
ncinanycase,aU
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832
GARRETT
BIRKHOFF
L5: If a C c, thena U (b
AND JOHN VON NEUMANN
n c) = (a U b) n c.
linearsubspacesof
Therefore(sincethe linearsum ofany two finite-dimensional
and consequentlyclosed) set-products
Hilbertspace is itselffinite-dimensional
and closedlinear sums of the finitedimensionalsubspaces of any topological
linearspace suchas Cartesiann-spaceor Hilbertspace satisfyL5, too.
One can interpretL5 directlyin various ways. First, it is evidentlya restrictedassociativelaw on mixedjoins and meets. It can equally well be regarded as a weakened distributivelaw, since if a C c, then a U (b n c) =
(a U c). And it is self-dual:
(a n c) u (bfnc) and (a U b) n c= (a U b)
replacingC, n, U by D, u, n merelyreplacesa, b,c, by c, b,a.
Also, speaking graphically,the assumptionthat a lattice L is "modular"
(i.e., satisfiesL5) is equivalent to24saying that L contains no sublattice isomorphicwiththelatticegraphedin fig.1:
n
FIG. 1
Thus in Hilbert space, one can finda counterexampleto L5 of this type.
Denote by ti, t2 t3 ..- a basis of orthonormalvectorsof the space, and by
a, b, and c respectivelythe closed linear subspaces generatedby the vectors
(Q2n + 1O-"tl + 10-2n2n+i), by the vectors 6,2n and by a and the vector ti.
Then a, b,and c generatethe latticeof Fig. 1.
Finally,themodularidentitycan be provedto be a consequenceof theassumpd(a), with the properties
tion that thereexistsa numericaldimension-function
D1: If a > b, thend(a) > d(b).
D2: d(a) + d(b) = d(an b) + d(a U b).
to latticesin whichthereis a
This theoremhas a converseunderthe restriction
? < a, < a2 < ... < a. < 0 of
finiteupper bound to the lengthn of chains25
elements.
Since conditionsD1-D2 partiallydescribe the formal propertiesof probability,the presenceof conditionL5 is closely related to the existenceof an
that any vector in (a U b) nc can be written t = a + , [a e a, , e b, t e c]. But = t- a
is in c (since t e c and a e a Cc); hence t = a +B e aU (bfnc), and aU (bf c) D (aUb) nc,
completing the proof.
24 R. Dedekind, "Werke," vol. 2, p. 255.
25 The statements of this paragraph are corollaries of Theorem 10.2 of G. Birkhoff,
op. cit.
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LOGIC
OF QUANTUM
MECHANICS
833
"a priorithermo-dynamic
weight of states." But it would be desirable to
propertiesofquantumphysics.
interpretL5 by simplerphenomenological
12. Relation to abstract projectivegeometries. We shall next investigate
how the assumptionof postulatesassertingthat the physicalqualities attributable to any quantum-mechanicalsysteme are a lattice satisfyingL5 and
L71-L73 characterizesthe resultingpropositionalcalculus. This question is
evidentlypurelyalgebraic.
We believe that the best way to findthis out is to introducean assumption
limitingthe lengthof chains of elements(assumptionof finitedimensions)of
thelattice,admittingfranklythattheassumptionis purelyheuristic.
L5 and L72 is the
that any latticeoffinitedimensionssatisfying
It is known26
directproductof a finitenumberof abstractprojectivegeometries(in the sense
ofVeblenand Young), and a finiteBoolean algebra,and conversely.
It is a corollarythat a latticesatisfyingL5 and L71-L73 possesses
REMARK:
independentbasic elementsofwhichany elementis a union,ifand onlyifit is a
Boolean algebra.
Again, such a lattice is a singleprojectivegeometryif and only if it is irreducible-that is, if and onlyif it contains no "neutral" elements.2Yx;S ?, 0
such thata = (a n x) u (a n x') forall a. In actual quantummechanicssuch
whichcommuteswithall projecan elementwould have a projection-operator,
ofobservables,and so withall operatorsofobservablesin general.
tion-operators
in quantummechanics.28
This wouldviolate the requirementof "irreducibility"
calculusofquantummechanicshas the
Hence we concludethat the propositional
as an abstract
geometry.
samestructure
projective
Moreover,this conclusionhas been obtained purelyby analyzinginternal
propertiesofthe calculus,in a way whichinvolvesHilbertspace onlyindirectly.
13. Abstractprojectivegeometriesand skew-fields. We shall now try to
get a freshpicture of the propositionalcalculus of quantum mechanics,by
recallingthe well-knowntwo-waycorrespondencebetweenabstract projective
geometriesand (not necessarilycommutative)fields.
and conNamely,let F be any suchfield,and considerthefollowingdefinitions
structions:n elementsxi, ... , Xnof F, notall = 0, forma right-ratio
[xi:. : Xn]r X
:
being called "equal," if and only
two right-ratios
[xi: ... Xn], and [.: ... .nl]r
ifa z e F with t, = xiz, j = 1, *.. , n, exists. Similarly,n elementsyi, *. , Yn
two left-ratios[yr: ... : Ynh
of F, not all = 0, forma left-ratio[yi: . .. :Y
and [1: ... 71,,] being called "equal," if and only if a z in F with ?li = zyi,
i = 1, ***, n, exists.
2r G. Birkhoff
"Combinatorial relations in projective geometries," Annals of Math. 36
(1935),743-8.
27 0. Ore,op. cit.,p. 419.
28 Usingthe terminology
offootnote,'and ofloc. cit. there: The ringMM' should conthan0, 1, or: the ringM mustbe a "factor." Cf. loc.
tain no otherprojection-operators
cit.'3,
p. 120.
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834
GARRETT
BIRKHOFF
AND JOHN VON NEUMANN
Now definean n - 1-dimensionalprojectivegeometryPn,-(F) as follows:
The "points" of Pn,-(F) are all right-ratios[x:...:xn]r.
The "linear subspaces" of Pmi-(F) are those sets of points,which are definedby systemsof
equations
ak1X1 +
+ akVXI, = 0,
k
=
1,*,m.
(m = 1, 2, *, theaki are fixed,but arbitraryelementsofF). The proof,that
this is an abstractprojectivegeometry,amountssimplyto restatingthe basic
propertiesoflineardependence.29
The same considerationsshow,that the (n - 2-dimensional)hyperplanesin
torn= 1,notall ai = 0. Put aii = yi,thenwe have
Pm-1(F) correspond
(*)
y1xl +
* + y.xn = O.
not all y = 0.
This proves,that the (n - 2-dimensional)hyperplanesin Pm._(F) are in a oneto-onecorrespondence
withthe left-ratios
[y.:... :Y.]
So we can identifythemwith the left-ratios,
as points are already identical
with the right-ratios,
and (*) becomes the definitionof "incidence" (point C
hyperplane).
Reciprocally,any abstractn - 1-dimensionalprojectivegeometryQn-,with
n = 4, 5, *-- belongsin this way to some (not necessarilycommutativefield
withPn..1(F(Qn..1))Y10
F(Qn1), and Qn-,is isomorphic
in
to involutory
14. Relation of abstractcomplementarity
anti-isomorphisms
skew-fields. We have seen that the familyof irreduciblelatticessatisfyingL5
and L72 is preciselythefamilyofprojectivegeometries,
providedwe excludethe
two-dimensionalcase. But what about L71 and L73? In other words, for
which P,1(F) can one definecomplementspossessingall the known formal
propertiesof orthogonalcomplements? The presentsection will be spent in
answeringthis question.3a
ofB. L. Van derWaerden's"ModerneAlgebra,"Berlin,1931,Vol. 2.
Cf. ??103-105
... means of course n - 1 2 3, that is, that Q._, is necessarily a "DesargueNew York,1910,
sian" geometry. (Cf. 0. Veblenand J.W. Young, "ProjectiveGeometry,"
in the classicalway. (Cf. Veblen
Vol. 1, page 41). Then F = F(Q-,,) can be constructed
betweenQni and the
and Young, Vol. 1, pages 141-150). The proofof the isomorphism
above,amountsto this: Introducing(not necessarilycommutative)
Pn,,(F) as constructed
theequationsofhypercoordinatesxi, **, x,,fromF in Q,,-,,and expressing
homogeneous
planes withtheirhelp. This can be done in the mannerwhichis familiarin projective
geometry,althoughmost books considerthe commutative("Pascalian") case only. D.
7th edition,1930,pages 96-103,considersthe nonHilbert,"Grundlagender Geometrie,"
and n - 1 = 2, 3 only.
geometry,
case, but foraffine
commutative
Consideringthe lengthyalthoughelementarycharacterof the completeproof,we propose to publishit elsewhere.
29
30n = 4, 5;
30a
R. Brauer, "A characterization of null systemsin projectivespace," Bull. Am. Math.
Soc. 42 (1936),247-54,treats the analogous questionin the oppositecase that X
? 0 is postulated.
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n x'
LOGIC
OF QUANTUM
835
MECHANICS
First,we shall show that it is sufficient
that F admit an involutory
antisomorphism
W: x = W(x), thatis:
Q1. w(w(u)) = U,
Q2. w(u + v) = w(u) + w(v),
Q3. w(uv) = w(v) w(u),
witha definite
diagonalHermitian
formw(xz1),yi +
+
*
W(xn)7n^,
where
Q4. w(x1),ylxl+ *** + w(x.)-yfx.= 0 impliesxi = ** = xn = 0,
the ey;beingfixedelementsofF, satisfying
w(yi) = ey.
Proof: Consider ennuples (not right- or left-ratios!)x: (xi,
,),
ofF. Defineforthemthevector-operations
) ofelements
Q1,
I:
**
xz: (XIz,
X +
t:
(XI
+
*.*
(z in F),
, xnz)
{i X
*so X
+
+
+
n
and an "innerproduct"
(Q1x)
=
W(e1)-YiX
W(fn)YnX.
Then thefollowingformulasare corollariesof Q1-Q4.
(x, 0) = w((Q,x)),
(Q,xu) = (Q,x)u, (eu, x) = w(u)( , x),
(Q, x' + x") = (Q, x') + (Q, x"), (Q' + V", x) = (i', x) + (W",
x),
(x, x) = w((x, x)) = [x] is # O ifx # 0 (that is, ifany xi #0 ).
IP1
IP2
IP3
IP4
We can definex 1 t (in words:"x is orthogonalto i") to mean that (%,x) = 0.
This is evidentlysymmetric
in x, t,and dependson the right-ratios
[xi:... :xn],,
... :
a .1 b,betweenthe
therelationof"polarity,"
[..
onlyso it establishes
points
b: [ti: ... : n]r ofPn.-(F).
a: [X1: ... :Xn]7,
]
: td] ofPn-i(F) constitutea linearsubspace
The polarsto anypointb: ...:
ofpointsofP,1(F), whichby Q4 does not containb itself,and yetwithb generates wholeprojectivespace P._,(F), sinceforany ennuplex: (xi, * X* , xn)
x =
X'
+
t.(]-1(QP,
x)
whereby Q4, [t] $ 0, and by IP (Q, x') = 0. This linearsubspaceis, therefore,
an n-2-dimensional
hyperplane.
elementofP.-,(F)1 one can set up inductively
Hence ifc is any k-dimensional
k mutuallypolar pointsb(l), ***, b(k)in c. Then it is easy to showthat the set
c' of pointspolar to everyb(l), ***, b(k)-or equivalentlyto everypoint in cconstitutean n-k-1-dimensional
element,satisfyingc nc' = ? and c U c' = 0.
Moreover,by symmetry(c')' D c, whence by dimensionalconsiderations
c -* c' defines
c" = c. Finally,c D d impliesc' C d', and so the correspondence
ofPnpl(F) completing
an involutory
theproof.
dual automorphism
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836
GARRETT BIRKHOFF AND JOHN VON NEUMANN
In the Appendixit will be shownthat thisconditionis also necessary. Thus
the above class of systemsis exactly the class of irreduciblelattices of finite
dimensions> 3 satisfyingL5 and L71-L73.
III.
CONCLUSIONS
15. Mathematicalmodels forpropositionalcalculi. One conclusion which
can be drawn fromthe preceding algebraic considerations,is that one can
models fora propositionalcalculus in quantum meconstructmany different
by knowncriteria. More precisely,one
chanics,whichcannotbe differentiated
satisfyingQ4 (such
can take any fieldF havingan involutoryanti-isomorphism
fieldsinclude the real, complex,and quaternionnumbersystems3l),introduce
and thenconstruct
suitablenotionsof lineardependenceand complementarity,
n a modelPn(F), havingall ofthe propertiesofthe
foreverydimension-number
propositionalcalculus suggestedby quantum-mechanics.
models P,(F) whose elements
One can also constructinfinite-dimensional
spaces. But
consistofall closedlinearsubspacesofnormedinfinite-dimensional
of
laws"
formal
(which
Hankel's principleof the "perseverance
philosophically,
of
analysis
technical
and
mathematically,
L5)32
leads one to try to preserve
continuous-dimensional
a
lead
one
to
prefer
in
Hilbert
space,
spectraltheory
modelPC(F), whichwillbe describedby one ofus in anotherpaper.33
PC(F) is veryanalogouswiththe modelfurnishedby the measurablesubsets
ofphase-spacein classicaldynamics.34
16. The logical coherence of quantum mechanics. The above heuristic
statementsin
suggestin particularthat thephysicallysignificant
considerations
quantummechanicsactuallyconstitutea sortof projectivegeometry,whilethe
a givensystemin classicaldynamics
statementsconcerning
physicallysignificant
constitutea Boolean algebra.
They suggesteven more stronglythat whereas in classical mechanics any
propositionalcalculusinvolvingmorethan two propositionscan be decomposed
into independentconstituents(direct sums in the sense of modern algebra),
quantum theoryinvolvesirreduciblepropositionalcalculi of unbounded complexity. This indicatesthat quantum mechanicshas a greaterlogicalcoherence
31
In therealcase,w(x) = x; in thecomplexcase,w(x + iy) = x - iy; in thequaternionic
case, w(u + ix + jy + kz) = u - ix - jy - kz; in all cases, the Xi are 1. Conversely, A.
Kolmogoroff,"Zur Begriindungder projektivenGeometrie,"Annals of Math. 33 (1932), 175-6
has shownthat any projectivegeometrywhose k-dimensionalelementshave a locally
relativeto whichthe latticeoperationsare continuous,must be over the
compacttopology
real,the complex,or the quaternionfield.
in P,(F) onlyelementswhich
by the artificeof considering
32 L5 can also be preserved
whichare offinitedimensions.
eitherare orhave complements
33 J. von Neumann,"Continuous
Proc. Nat. Acad., 22 (1936),92-100and
geometries,"
101-109. These maybe a moresuitableframeforquantumtheory,thanHilbertspace.
34 In quantummechanics,dimensions
are uniquelydeterminedby
but not complements
the inclusionrelation;in classical mechanics,the reverseis true!
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LOGIC
OF QUANTUM
MECHANICS
837
than classical mechanics-a conclusioncorroboratedby the impossibilityin
quantitiesindependently.
generalofmeasuringdifferent
17. Relation to pure logic. The models for propositionalcalculi which
have been consideredin the precedingsections are also interestingfromthe
standpointof pure logic. Their nature is determinedby quasi-physicaland
fromtheintrospective
and philosophicalconsideratechnicalreasoning,different
to comtionswhichhave had to guidelogicianshitherto. Hence it is interesting
pare the modificationswhichthey introduceinto Boolean algebra, with those
whichlogicianson "intuitionist"and relatedgroundshave triedintroducing.
seems to be that whereaslogicianshave usuallyassumed
The main difference
that propertiesL71-L73 of negationwere the ones least able to withstanda
identitiesL6 as
criticalanalysis,the studyof mechanicspointsto the distributive
theweakestlinkin thealgebraoflogic. Cf. thelast two paragraphsof ?10.
Our conclusionagrees perhapsmorewith those critiquesof logic,whichfind
most objectionablethe assumptionthat a' U b = 0 impliesa C b (or dually,
the assumptionthat a n b' = ? impliesb D a-the assumptionthat to deduce
that
an absurdityfromthe conjunctionof a and not b, justifiesone in inferring
a impliesb).36
18. Suggested questions. The same heuristicreasoningsuggeststhe followquestions.
ing as fruitful
What experimentalmeaningcan one attach to the meetand join of two given
experimentalpropositions?
What simpleand plausiblephysicalmotivationis thereforconditionL5?
APPENDIX
1. Considera projectivegeometryQ,,i as describedin ?13. F is a (not necessarilycommutative,but associative) field,n = 4, 5, ... , Q.-, = PnA.(F)the
projective geometryof all right-ratios[xI: * :xnlI, which are the points of
are representedby the left-ratios
Qn-1. The (n - 2-dimensional)hyperplanes
[yI: ... Yn1 incidenceof a point [xI:... xn], and of a hyperplane[Yi:... Yn1l
beingdefinedby
n
(1)
z yixi
= =0
1YXi
All linearsubspacesofQ.-, formthelatticeL, withtheelementsa, b,c, *ii
Assumenowthatan operationa' withthepropertiesL71-L73 in ?9 exists:
L71 (a')' = a
=?
L72 ana'
and a U a' = O,
L73 a C b impliesa' D b'.
35
law
to show,thatassuming
ouraxiomsL1-5and7, thedistributive
It is notdifficult
L6 is equivalentto thispostulate:a' U b = Uimpliesa C b.
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838
AND JOHN VON NEUMANN
BIRKHOFF
GARRETT
They imply(cf. ?9)
(afn b)' = a' U b' and (a U b)' = a'
L74
f b'.
in a, b,owingto L73 and L71.
Observe,thattherelationa C b' is symmetric
2. If a: [x1:... : x,],is a point,thena' is an [yr:... yJh. So we maywrite:
[x:l*** Xn1r,=
(2)
[lyi .. ** s]l
^
and definean operationwhichconnectsright-and left-ratios. We know from
?14, that a generalcharacterizationof a' (a any elementof L) is obtained,as
soon as we derive an algebraic characterizationof the above [xi:*...* : x]r .
We will now findsuch a characterizationof [xi: * :x"] , and show, that it
justifiesthedescriptiongivenin ?14.
in
In orderto do this,we will have to make a ratherfreeuse of collineation3
which
Qn-1. A collineation is, by definition,a coordinate-transformation,
replaces[xi: ** :xn]4 by [xl: ... **n]. ,
(3)
n
xi=
i -2
Xi
aii
forj = 1p -n.
Here the wijare fixedelementsofF, and such,that (3) has an inverse.
(4)
n
xi=
j=1
Oiji,
fori =1,
,n,
theOijbeingfixedelementsofF, too. (3), (4) clearlymean
lifk = 1
8k1 =
(5)
n
s2iOtj
j=1
0
~
11~
(ki
ifk 5e 1
ik
Sik
J
n1
(j Oik =
i-1
jik=
Considering(1) and (5) theyimplythe contravariantco6rdinate-transformation
forhyperplanes:[yi: ... :yn],becomes[pi: ... :9,]j, where
Yi=
(6)
yi=
(7)
n
i 21
n
j= 1
Yioii,
forj = 1,-
aii
n,
fori= 1,***,n.
on the leftside of the variablesin
(Observe,that the positionof the coefficients
(4), (5), and on theirrightside in (6), (7), is essential!)
3. We willbringabout
(8)
[dil:Si
n**R]'
=
[MI:...
*
in~l
fori
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=
1, *
n,
LOGIC
839
MECHANICS
OF QUANTUM
by choosinga suitablesystemof coordinates,that is, by applyingsuitable collineations. We proceedby induction:Assume that (8) holds fori = 1,
m - 1(m = 1, * , n), then we shall finda collineation which makes (8) true for
i= 1,.***,m.
Denote the point [5il: ..*. :sn]r by pi, and the hyperplane[6il: ... : 65in by
h*our assumptionon (8) is: p*' = h*fori = 1,.**, mr-1. Considernow a
point a: [xl:* *:x,],, and the hyperplanea': [yi:...
:y** ]i. Now a _ p*' = h*
means (use (1)) x, = 0 and p* < a' means (use (8)) y; = 0. But thesetwo
statements are equivalent. So we see: If i = 1, * , m - 1, then xi = 0 and
y, = 0 are equivalent.
As 8mS= 0 for
Put Pm: [y*:.. . :y].
Consider now p*: [3mi:...:*mnnr.
, m-1.
Furthermore,
i = 1, ***,m-1,
so we have y =0 fori = 1,
p f pea o, p* . o, so p*not _ p*'. By (1) thismeans y#P O.
Formthecollineation(3), (4), (6), (7), with
...
-
09si= col;
1sV
im =
@swi =
YM
for i = m + 1, *;,n,
Y*
all other0,s,ws = 0.
One verifiesimmediately,that thiscollineationleaves the coordinatesof the
n, invariant, and similarlythose of the
Pi: [5i:l *
in], i =
1, :5iin]i, i = 1, ***m - 1, whileit transformsthoseof
p*: [aj:.
into [8i:
...* 5*M]l
So after this collineation (8) holds fori = 1, **, m.
Thus we may assume, by induction over m = 1, ** , n, that (8) holds for all
i = 1, ... , n. This we willdo.
The above argumentnow shows,that fora:
[xi:
...
xi = 0 is equivalent to yi = 0,
(9)
4. Put a:
[xi: ...- :xlr,
a':
[yi: ...
:yn]z,
further xi =
1.
Then
(9) gives
*y"]1
for i = 1, ... , n.
:
and b: [i: ... -n]rb':
Assume
first i1 = 1, X2 = 17117=3
?I=
= ** =
we can normalize ti = 1, and
Only, so tb = f2(77).
Assume
a': [yr:...
:xn]r,
n = 0.
n = 0.
yj i
1n]
[771
i 5= O, so
{a can depend on 712 =
Then
(9)
gives
0, so we can normalize
Yi =
1.
= 0.
Now a S b' means by (i) 1 + IX2 = 0, and b S a' means 1 + Y2f2(G)
So if x2 # 0, we may put
These two statements must, therefore, be equivalent.
If x2 = 0, then
=
=-(f2(1i, and obtain Y2 = fX-'))-f'
2(f))-'
of
0 by (9).
Thus, x2 determines at any rate y2 (independently
y2=
Permuting the i = 2, * * *, n gives, therefore:
Xa,
Xn) : Y2 = (P2(X2).
There exists for each i = 2, ... , n a function pi(x), such that y; = (ii)
*
*
Or:
(10)
If
a:
[1: XI ... : Xn]r,
then
a':
[1:V2(x2):
...* :n(Xn)]l-
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840
GARRETT
Applying this to a: [1:X2:
and c < a' are equivalent,so
(11)
AND JOHN VON NEUMANN
BIRKHOFF
...:
X]r and c: [1: u1:...
n
:u
shows: As a < c'
n
E Vi(ui)xi = -1 is equivalentto E jp(xi)vi =-1.
i -2
i
2
that(9) becomes:
Observe,
(Pi(x)= Oifand onlyifx = 0.
(12)
5. (11) withX3=
Xn =
=
*..
is equivalentto IP2(X2)U2
=
-
1.
equationholds,and so bothdo.
Choose X2,
=
U3
If X2
U2 in thisway, butleave x3,
0, u2
...
(11) becomes:
n
(13)
E
i-3
Nowputx5 =
i-3
=
=n*=
=
s.Un
0 is
0.
shows:IP2(U2)X2
(-IP2(X2))-f,
Xn, U3,
n
P3(U3)X3 + io4(u4)X4 =
thatis (forX4,U4
5s
,
=
(pi(ui) xi = 0 is equivalentto E
.*
0
n =
=
*..
z-
...
= - 1
thenthe second
unarbitrary. Then
(pi(xi)Ui = 0.
Then(13) becomes:
equivalentto IP3(X3)U3
+ IP4(X4)U4 =
0,
0):
(a)
x3x7l = so4(U4)-1
'93(ug)
(b)
u3u-1
to
(14) is equivalent
4(x4) -1 v(x8).
=
Let x4, X3 be given. Choose u3, U4so as to satisfy(b). Then (a) is true,too.
Now (a) remainstrue,if we leave U3,
U4unchanged,but changex, z4,without
thatis, the
changing
x3x41. So (b) remainstoo trueundertheseconditions,
value of P4(x4)-1V&(X3) does not change. In otherwords:(P4(X4)- sP3(x3)depends
on x3x-1only. That is: (P4(X4)-1 V3(X3) = (P34(X3X-4). Put X3 = XZ,X4 = x,
thenwe obtain:
(15)
(P3(xz) = IP4(X) {134(Z).
Thiswas derivedforx, z $ 0, butit willholdforx or z = 0, too,ifwe define
134(0) = 0. (Use (12).)
(15), withz = 1 givesIP3(x) = J04(X)a34,where a34 = 4,34(1) 0 0, owing to
thei = 2, * , n gives,therefore:
(12) forx 0 0. Permuting
(16)
Vi(x) = ros(x)ai,,
(Fori =jput aii = 1.)
Now (15) becomes
(17)
W((Z)
=
(P2(ZX)
=
whereaij 0 0.
(P2(X)w(z)
a4234'(z)a23.
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LOGIC
OF QUANTUM
841
MECHANICS
Put x = 1 in (17), writex forz, and use (16) withj = 2:
(Ps(x) =
(18)
#w(z)yi,
(O =
6. Compare(17) forx = 1, z=
Then
whereA, yj F 0.
IP2(1)l -yi =
ai2).
u;x=uz=v;andx=1,z=vu.
w(vu) = w(u)w(v)
(19)
results(12) and (18) give
w(u) = 0 if and only if u = 0.
(20)
Now writew(z), ,y for,w(z)t-', jBy. Then (18), (19), (20) remaintrue,(18)
in so far,as we have i = 1 there. So (11) becomes
is simplified
n
Ew(Ui) 'Yix = -
(21)
i =2
(21) is equivalentto
n
E
i -2
W(Xi) jiUi =-1
u and all otherxi = ui = 0 give: w(u)Y2x =-1
is equivalentto
If x # 0, u = -72 1 w(x)-', thenthesecondequationholds,
But
and so the firstone gives: x =-y1
w(u)-1 =-2 -y(w(--yj'w(x)-1))-l.
(19), (20) implyw(1) = 1, w(w-')
w(w)-', so the above relationbecomes:
X2 =
x, u2
w(x)'yu
=
=
-1.
-
X = -n'(w(-8^
=
= -2
-w(x71))-
-y:'W(W(X)(-Y2))
x = 1,as w(w(1))
Put herein
=-72
=
W(1)
=
lW((-8-l(X)-l)-l)
W(-2)W(W(X)).
1, SO -'2y
results. Thus the above equation becomes
(22)
and w(-Y2)
=
1W(-Y2)
=-2
w(w(x)) = x,
=
7-2
gives,ifwe permutethei = 2,
Put ui =-'yr'
n,
w(-'yi) = -Fi.
(23)
(24)
W(--Y2)
in (21).
n
E
. -2
Then considering(22) and (19)
n
xi = 1 is equivalentto 2: w(xi) = 1
i =2
*
= n =O.
Then(24)
obtains.Put x2= x, x3 = YX4=1-X-yX6=
gives w(x) + w(y) = 1- w(1 - xy). So w(x) + w(y) dependson x + y
only. Replacingx, y by x + y,0 shows,that it is equal to w(x + y) + w(O) =
w(x + y) (use 20). So we have:
(25)
w(x) + w(y) = w(x + y)
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842
GARRETT
BIRKHOFF
AND JOHN VON NEUMANN
(25), (19) and (22) give together:
ofF.
antisomorphi8m
w(x) is an involutory
Observe,that (25) impliesw(-1) = -w(l)
w('yi) =
(26)
=-1,
and so (23) becomes
'vi.
7. Consider a: [x1:... : Xz]r, a': [yi:... : yn]. If xi 5 0, we may write
a: [1:x2x': ... :xfx1],r,and so a': [1:w(x2X')-Y2: ... :w(nx71)'Y,],. But
Yi = W(Xl)-Iw(xi)-y;,
w(xi~xT)'vi= W(XT1)W(xi)
and so we can write
a': [w(x1):W(X2)*2: ...
W(X:)-Ynl
too. So we have
fori = 1, ...
yi = w(xi)'vi
(27)
n,
wherethe -Yifori = 2, * , n are those from6., and 'vI = 1. And w(1) = 1,
(27) with'vj
so (26) holdsforall i = 1, * , n. So we have the representation
0.
obeying(26), ifxi 5
Permutationofthei =1, *... , n shows,thata similarrelationholdsifx2 5 0:
yi = w+(xi)iT+,
(27+)
'vit
W+(uS) =
(26+)
of F. (w+(x), -ytmay differfrom
w+(x) being an involutoryantisomorphism
now
1
have
we
=
of
w(x), Syi!) Instead -yv
y+ = 1, but we willnotuse this.
=
a':
Then
[yr:
1.
all
Put
.*:y,]z can be expressedby both formulae
xi
so w(1) = w+(1) = 1,
(27) and (27+). As w(x)lw+(x) are both antisomorphism,
...**zl
: :*
obtains. Thus
['=
and therefore[yi: ... :y]l = [-I: ... *:
(-I+l)-1-
(yi)-f i = 'i,'i+
=
-
ye+7ifor i = 1,
...
, n.
but as we are
Assume now x2 5 0 only. Then (27+) gives yi = w+(xi)'y+i,
dealingwithleftratios,we mayas wellput
Yi = (+4)-1w+(xi)yt = ('1+) Yw+(x)'y+ty,.
Put
=
'v
9+ 0, thenwe have:
ei
yi (xi)-+ W+
(27++)
Put now xi = X2 = 1, X8 = X, all otherxi = 0. Againa': [yi: ...* yn]lcan be
expressedby both formulae(27) and (27++), again w(1) = w+(1). Therefore
[y1
y2: y
y4:...
:yn1l
=
[l:-Y2:W(X)-Y3:0
:...
?]1
=
[v1:'2:t+-W+(X)f'+Y:0:
..
?]1
obtains. This impliesw(x) = ft+w(x)t+ forall x, and so (27 ) coincideswith
(27).
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LOGIC
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MECHANICS
843
In otherwords: (27) holds forx2 z- 0 too.
Permutingi = 2, ***, n (onlyi = 1 has an exceptionalr6lein (27)), we see:
(27) holds if xi 5 0 fori = 2, * , n. For xlI5 0 (27) heldanyhow,and for
somei = 1, ... , n we musthave xi 0O. Therefore:
(27) holdsforall pointsa: [xi: *-:- x4],.
8. Consider now two points a: [xi: ... :x4] and b: [ti - n.. Put
Xthenb ? a' means,considering(1) and (27) (cf. the end of 7.):
a'i:[y.... : yn,1
n
(28)
E
i-1
a ? a' can neverhold (a
ifall xi = O. Thus,
(29)
u(xi)yi(i = 0.
n a' = 0, a
# 0), so (28) can only hold forxi = (i,
n
= 0 impliesxi =
Ad w(x1),yix
i-1
*-- = Xn = 0.
Summingup the last resultof6., and formulae(26), (29) and (28), we obtain:
Thereexistsan involuitory
w(x) ofF (cf. (22), (25), (19)) and a
antisomorphism
n
definite
diagonalHermitianform E w(xi)yi i in F (cf. (26), (29)), suchthatfor
a: [xl:*.
(28)
b: [i:
:Xnl]7,
.*.
i-1
] b < a' is defined
bypolaritywithrespecttoit:
n
A w(xi)yi{i = 0.
This is exactlythe resultof ?14,whichis thusjustified.
THE SOCIETY OF FELLOWS, HARVARD UNIVERSITY.
TEns INSTITUT FOR ADVANCED STUDY.
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