PoS(FFP14)207 Philosophy of http://pos.sissa.it/ 263523, Project ◦ ce. g the reduction procedure used in in claim of this paper is that this describe a physical system can be antum states are defined by half the eduction. This fact points towards a - CNRS, Paris, France. eille, France inciple in quantum mechanics. Filosofía y Letras, Universidad de rom the European Research Council under the Euro- 013 Grant Agreement n completely ive Commons Attribution-NonCommercial-ShareAlike Licen ). † ∗ [email protected] gauge theories, namely the Mardsen-Weinstein symplectic r understood as a consequence of the same formalism underlyin gauge-theoretical interpretation of the indeterminacy pr number of variables required in“reduction” classical in mechanics. the number of The variables ma required to As a consequence of Heisenberg indeterminacy principle, qu Speaker. The research leading to these results has received funding f ∗ † Copyright owned by the author(s) under the terms of the Creat c Frontiers of Fundamental Physics 14 15-18 July 2014 Aix Marseille University (AMU) Saint-Charles Campus, Mars 1- Laboratoire SPHERE (UMR 7219), Université2- Paris Instituto Diderot de Filosofía “Dr. Alejandro Korn”, Facultad de Buenos Aires - CONICET, Buenos Aires,E-mail: Argentina. pean Community’s Seventh Framework Programme (FP7/2007-2

Canonical Quantum Gravity Gabriel Catren Quantum Indeterminacy, Gauge Symmetries and Symplectic Reduction PoS(FFP14)207 q (2.1) observ- is com- degrees reduced i n i p . As it is the | Gabriel Catren observables ω –i.e. the so-called [1]. The symplectic constrained Hamilto- ω preserving n classical quantum variables can be n G n the amount of (invariant) in- , far from being a characteristic it important mathematical details underlying the gauge symmetries, i i chanics (see Refs.[2, 3, 4] for a dis- mplemented at the infinitesimal level tion from 2 act that quantum states are defined by re general states (such as for instance classical operators p he number of observables required to uch a reduction. In the simplest case, m. One of the main examples of this s. Heisenberg indeterminacy principle racterized by a well-defined value of the rom being a mere analogy, the quantum changing the position without affecting duction (i.e. the invariance modulo a phase factor) i ≈ | i –and q p | M degrees of freedom. Now, the transition from i , p f 0 classical variables to Classical operators v on iq of the system described by the state 2 n is provided by the theory of π ) 2 q → 7→ e M ( f (see also Refs.[2, 3, 4]). In what follows, we shall fo- physical ∞ . The fact that quantum states are defined modulo phase ) i 7→ C k i operators acting on states p ∈ − reduction · | f n q or p 0 ( and q –. It is worth stressing that this correspondence between and Observables first-class constraints generating gauge symmetries can be M ) [7]. In these theories, a classical system described by 2 : is endowed with an action of a Lie group k ω on M f observables symmetries under translations in the position –i.e. functions of the momentum operator i i p gauge theories | (or n quantum permits to define a map symplectic reduction procedure be a connected manifold endowed with a symplectic structure ω observables M to the Let In general, the presence of symmetries entails a reduction i Let’s suppose now that phase invariance classical to quantum mechanics alsodefine entails a a physical reduction state, in namely t (in the simplest case) a reduc understood as a consequence ofnamely the the same geometric formalism cus on the conceptual aspects(a of more the detailed arguments presentation and can we be found shall in om Ref.[5]). 2. The Moment Map can be interpreted by saying that the position Quantum Indeterminacy, Gauge Symmetries and Symplectic Re 1. Introduction formation required to completely describerelationship between a physical syste nian systems between ables assigning numbers to states feature of quantum mechanics, iscussion at of the this point). heart of classical me structure case with every group action on a manifold, such an action is i to a physical system described by 2 the state can be rephrased by sayingposition. that the In latter this is way, quantum not phase cha symmetries encode the f pletely “undetermined”. In other terms, the possibility of half the number of variablescan required be in understood classical as mechanic a generalizationcoherent of states). this The reduction main to stance mo ofphase this symmetries paper encoding the is reduction that, from far 2 f and p of freedom and endowed with Hamiltonian vector fields v of an eigenstate factors can be understood as a particular manifestation of s the PoS(FFP14)207 ) g M ( . In (2.3) (2.2) q ∞ of the ∗ C tion via g ) such that ouriau [11]. g and ng function. . This map is Gabriel Catren g . The action is X M respondence (2.1), on generating function terms? One possible X -action will be called unitary representation v G ap. Both . On the other hand, the R associated to → , the co-moment map can be onian such that the classical operator ∗ M g X f and elements in the dual , are dual in the sense that there is respectively. On the one hand, tails the existence of a privileged × . In turn, the co-moment map is on i ) p ) . Briefly, the action is Hamiltonian g servable (called M X . By means of these dualities we can M by translations in the position [8]. According to this method (which : . More precisely, there is a map that m v M ) ω ( ( } 7→ m ·i ) is Hamiltonian there is a relationship– µ ∞ duction ( , ) , and p f ) p C h· , X ∗ , M q h M g q ∗ ( ( 7→ ( g { ∞ ) : )= m C = → µ , m 3 . The simplest example of this formalism is given f , namely the observables given by the generating ( ) M M to a fundamental vector field ( → X M : f G , G g on µ : ω encodes (what we could call) the of , ˜ )= R µ to a generating function ∗ g M m g ( g = Kirillov’s orbit method )( given by ) ), ∈ G X fundamental vector fields R G ( –between classical states in X is the coordinate of the 1-dimensional vector space ˜ µ µ → 0 . The classical operator associated to this generating func q p M 0 called q × M and denoted ) = that generate the group action can be derived from a generati and the algebra of observables moment map M and was introduced by the French mathematician Jean-Marie S i (where ( ) M in the Lie algebra M 0 p . ∞ q , X if these fundamental vector fields can be obtained via the cor f . C M q q ( ∂ ∂ ∈ 7→ µ 0 , q 0 m 0 . Now, how can we understand this relationship in conceptual q q g = : h 0 and ˜ q µ f g v )= are dual in the sense that there is a bilinear map co-moment map Hamiltonian p ∈ ∗ , When a group action on a symplectic manifold In what follows, a symplectic manifold endowed with a Hamilt In what follows we shall also need a map dual to the co-moment m g q X ( 0 q f 3. Kirillov's Orbit Method and Symplectic Reduction established by the moment map which is dual to therecovered by co-moment means map of (2.2). the following Given expression the moment map (2.1) is Quantum Indeterminacy, Gauge Symmetries and Symplectic Re by some vector fields on defined by the latter via (2.1) is the fundamental vector field called if the vector fields on only works for certain Lie groups called sending each Lie algebra element given by by the action of the abelian group for sends each element This means that given a Hamiltonian action we can define a map Hamiltonian G-manifold In this way, a Hamiltoniansubalgebra action of on the a algebra symplecticfunctions of manifold of observables the en on group action. this case, the moment map is simply given by introduce a map (called and i.e. if each fundamental vector field can be derived from an ob of the group action) by means of the map (2.1) defined by in (2.2) have dual structures, namely the linear dual Lie algebra an evaluation map answer to this question is provided by symplectic manifold PoS(FFP14)207 . . ∗ G . M G g The G of (3.2) (3.1) in 1 is that . ξ ∗ ρ ect sum G O classical in short) g leaves its ) described ∗ g O Gabriel Catren . However, with respect to , we expect the G ∗ action of g . In this case, the unirreps G can be decomposed as G-irreducible carrying unirreps of to be irreducible. Now, all denote -action on . O G M -irreducible classical sys- G -action on these vectors is i irreducible ξ H transitive G obtained by quantizing the G | H O Lie group are ∗ . In other terms, each classical H ∗ g can be obtained by quantizing an . g representations ( g in G yields a 1-dimensional Hilbert space ∈ tization of the e foliated in symplectic manifolds. In esulting quantum theory is composed ∗ , ). Secondly, it is a well-known fact in abelian O carried by g X is not necessarily transitive. Therefore, O defines a 1-dimensional unirrep duction has the structure of a Poisson manifold G ∈ ∗ H M ) g ∗ ξ can always be connected by means of a group element g . Let’s consider now the quantization of the , M ∈ on , irreducible defined by the unirrep (3.1). If we assume that and Hilbert spaces O O G , , is endowed with a ξ i ∗ ) ∗ G O encodes the unitary representation theory of H X ( 1 g g , 4 ( ∗ ξ ) m h carrying a unitary representation of g in

i ∗ 1 U g π ( M 2 O coadjoint action ⊂ O -invariant subspaces within a classical system e U M O H G = → 7→ -action on M X G can be understood as a family of G . In other terms, we have a decomposition of the form e : . Since any unitary representation of are just points. In other terms, the ∗ ∗ H quantum state that we shall denote ∗ g g ξ ∗ g ρ g in . Now, the remarkable fact regarding the Poisson manifold (the so-called can be decomposed (as every unitary representation) as a dir -class of normalized vectors. The unitary ∗ ) O g . Analogously to the case of the classical systems in G 1 M ( M H U , i.e. as a direct sum of the Hilbert spaces -irreducible, i.e. the action of -action means that the classical systems -action are exactly the symplectic leaves of G to yield a Hilbert space -phase invariant G G G ) M ξ -manifold , being an orbit of the , -irreducible classical systems ∗ can be understood as a family of classical systems (that we sh G G g G . Let’s unpack this statement. Firstly, ∗ ( yields quantum systems that are unitary g G in O O of -action. In other terms, there are no The transitivity of the action means that any two points in We have thus far argued that The quantization of the one-point classical system In what follows, we shall only consider the case of an . In other terms, the process of quantization defines a map containing a unique 1 G . G ξ G -irreducible classical systems in systems between the points invariant, i.e. each point is an orbit. A point by the symplectic leaves of of a unique quantum states are define modulo overall phase factors, the r irreducible classical systems in in tems encoding the unitary representation theory of In the cases for which the orbit method works, any unirrep of Hamiltonian of implemented by means of the phase factors in given by Quantum Indeterminacy, Gauge Symmetries and Symplectic Re theory irreducible classical system the orbits of the the representation space of the unirreps of system the theory of Poisson manifolds that aparticular, Poisson manifold can b transitivity of the a direct sum of unirreps, the Poisson manifold there is no reason to expect the unitary representation of H G the endowed with an action of quantization of is not necessarily Now, the main idea of Kirillov’s orbit method is that the quan PoS(FFP14)207 , - ) µ G . G , M ω occurs H , (and the ξ M procedure ( M H 0 This/0. is the H of the theory -equivariant– Gabriel Catren G is a symplectic 6= intertwining the is not in general that occur in the yields the multi- occurs in ξ G satisfying the fol- ) ξ M M -orbits would give ξ / O . Now, how can we ξ defined by the value ) ξ o be would yield the term is a Hamiltonian G ( H M H H -manifold -symplectic reduction ξ M M H 1 ) ( ) M ξ G − if 1 in (3.2). However, this ξ H that the moment map → G ep ( µ − ⊂ , ξ 1 M ξ that occur in ) µ gauge group M -space, i.e. a collection of − ω 0 H , H H ( . 1, i.e. the unirrep µ G G ) n = 1 . M ion of this result). This implies − ξ M ( is a one-point space, the Hilbert . Let’s describe how this works -manifold , trate it for a particular case) that )= µ M ollection of is a µ n of ξ ξ reimage G ( ed. Now, a fundamental result in M ) M , m ξ will be called ξ ( occurs in ( tient ξ , i.e. the subset 1 –where is called the is called a gauge theory if the Hamilton Sternberg conjectured that the moment ξ m − yields the unirreps ξ duction M ) M a is that we can construct the decomposi- µ G H M G . to -symplectic quotients H , µ ξ M M ω by means of . In this way, whereas the quantization of the , yields a Hilbert space , expression (3.2) means that the multiplicities ) ⊂ M O ξ ) M -symplectic quotient 5 M ( , ξ constraint surface M ξ Mardsen-Weinstein symplectic reduction H ( to (i.e. its dimension) gives the number of independent . In particular, if ξ associated to a Hamiltonian 1 ( ξ µ − O defined by the (one-point) irreducible classical system m M µ ξ H H –, the preimage H ) defines a map of Hilbert spaces m ( ξ µ M · will be called 1 H gauge theory ξ − M g -orbit, thereby yielding the term , the quantization of the belonging to the image of is in the image of G M -symplectic quotient ξ )= ξ H ξ m · , i.e. the multiplicity g M ( µ for each encodes the information concerning the unirreps of . . Therefore, the 1-dimensional unirrep H is the number of times (i.e. the multiplicity) that the unirr ) ∗ M ξ ) g in M 0 that/0, is if H H , in the decomposition (3.2). Since the moment map is assumed t M i is composed of a unique quantum state. Then, , ξ → ξ ξ 6= ( | ξ completely define the quantization of the Hamiltonian O -action on both spaces (see Ref.[5] for a conceptual discuss is not necessarily abelian. A theory on ξ M M m ( H ) G ) : of the moment map [9]. In this case, the group M G . To do so, we have to consider the subset of classical states i m H M ∗ ∗ M µ , If we assume that we know the unirreps We shall now consider a In a first rough approach, we could guess that the quantizatio Guillemin and Sternberg conjectured in Ref.[6] (and demons g , g O ξ ∈ ∈ ( ( -orbits. We could then guess that the quantization of these c m calculate these multiplicities? Inmap Ref.[6], Guillemin and unitary manifold. This is the content[10]. of In what the follows, so-called the procedure of passing from a symplectic manifold, whichsymplectic means geometry that is it that, cannot under nice be conditions, quantiz the quo ξ puts in correspondence to the irreducible classical system Quantum Indeterminacy, Gauge Symmetries and Symplectic Re where equations “constraint” the solutions to be in the corresponding multiplicities). Roughly speaking, thetion ide (3.2) by “pulling-back” the unirreps 4. Quantization Commutes with Reduction and the symplectic manifold decomposition (3.2) of space only once in in the abelian case by calculating the multiplicity in m idea does not work for a fundamental reason, namely that the p lowing property: each state in copies of (trivial) classical system plicities that the number of independent states in where 0 the quantization of the case if manifold–of the 1-dimensional unirrep G one copy of in the sense that PoS(FFP14)207 : - is ly gau a for are G gau ϕ µ M (4.1) (4.2) (4.3) . We M G . This H H M -invariant . We can defined by M G -invariance 0 Gabriel Catren H . In this case, M gau G G defines a map Dirac’s method ) defined by the co- 0 tion g must be a M ing that “quantization ∈ is the quantum operator M i H and then select the i X ˆ f implements the notion of H unirrep of M M in (for yields a 1-dimensional Hilbert a H i ∗ gau i ϕ part of the classical symplectic -invariant states where (which is in general problematic) g trivial -invariant states in G X f the so-called M , , gau G ∈ ) . Diagramatically, 0 M G H -invariant quantum states in M -invariant quantum states on O  O in a basis of O M m defining the trivial unirrep of G f (4.3) guarantees that one can either = ( ce a ≃ H gau 0 gau H / µ duction 0 ψ ψ h G i /o/o/o/o G ˆ M f , gau H )= G / 0. The Mardsen-Weinstein reduction theorem gau /o/o / G m ) M such that the corresponding moment map ) carrying the ( /o/o/o/o i 0 i )= f ( /o/o/o 6 H 0 . In other terms, we want to consider symplectic 1 | m ) theories (i.e. theories that are not gauge theories) ( − ≃ i G -action, the image of µ 0 , /o/o /o/o Quantization -action M G ω ≃ , G afterwards or directly quantize 0 H 0 M 0 O  O is in bijective correspondence with the Hilbert space O O M ( -invariant quantum states in M ordinary M M ∗ G g . The reason is that the constraint surface can be equivalent obtained by quantizing the 0-symplectic quotient defines a set of independent (Dirac’s method). The ∈ in the physics literature) is a symplectic manifold. 0 0 M M M -invariant quantum states in [7]. The point that we want to stress here is that the H H i H f gau -manifolds G quantum constraint equations G constraints constraint equations f ) and the generating functions -symplectic reduction gau 0 denotes the Hilbert space generated by the G at the quantum level. Therefore, we can rephrase (4.2) by say reduced phase space gau . Since this map intertwines the with respect to the irreducible classical system G M M H H Now, we are here interested in Now, the quantization of the irreducible classical system 0 The commutativity of this diagram guarantees the validity o i→ -reduction 0 defined by means of the moment map are called Quantum Indeterminacy, Gauge Symmetries and Symplectic Re (and denoted commutes with 0-reduction”. 5. On the Quantum Indeterminacy as a Form of Symplectic Reduc states that the 0-symplectic quotient | (also called manifolds endowed with a Hamiltonian state. All in all, a basis of defined by Hamiltonian associated to the constraint space (whose unique quantum state is denoted rephrase this result by means of the following isomorphism where could say that the selection of the obtained by selecting the of the quantum statesreduction of a gauge theory is the quantum counter 0 quantizing gauge theories [7].reduce Indeed, the classical the theory commutativity defined o by the original phase spa and quantize the reduced phase space means that the Hilbert space the irreducible classical system 0 the Guillemin-Sternberg conjecture states that each state invariants quantum states in the states that satisfy the PoS(FFP14)207 , , M gau and H (5.1) (5.2) (5.3) G do we gram” portant in M quantum -invariant H ), the phase G . In the case ξ Gabriel Catren gau art of the sym- M G phase group of the moment map i defined by the 1-dim. p ) 1 ( -phase invariant quantum ) U ξ , does not yield the will be called G -phase invariant states ( ) containing a unique point. ξ . G . Therefore, while the action of a ) . ξ tization of the unique symplectic . Differently from the constraints M ξ , G g G -phase invariant states , , each value ) / G G ξ q ) ociated to the trivial unirrep of ∈ ( , -symplectic quotients ( i M -reduction” provided that the notion of G p ξ X ( M ( O  O O ξ H irrep of the gauge group duction ing isomorphism (see Ref.[5] for details) 1 . Now, what is the analogue of this bijec- riance appearing in gauge theories to the H − G . ≃ / M is supposed to be abelian). In this way, the ) / /o/o µ ξ ξ ) , H of the moment map. This means that ? It can be shown that in the abelian case the G M . ξ G = ξ , ( ( /o/o/o/o ξ i i M 1 H p M X / , ? In other terms, what kind of states in − 7 /o H -symplectic quotient ξ ) ξ M h µ i /o/o ξ 1 phase observables ≃ M π ( . (where value i 2 = /o/o/o/o f ξ e U /o/o by translations in 0. ξ G -symplectic quotient M } ξ M 6= ) H Quantization → 7→ /o/o . In these cases, the group p ξ ∗ , /o X G g e q : non-zero ( ξ O O  O O { ξ M -phase invariant states. The corresponding commuting “dia M -invariant states as it is the case in gauge theories). The im ρ ) = ξ gau , : M G G ξ ( -symplectic quotients -symplectic quotient ξ i on -invariant quantum states in p -symplectic quotients of the phase group R -phase invariance of quantum states is the quantum counterp G ) ξ entails the existence of a whole set of g that are invariant modulo overall phase factors in ξ , denotes the Hilbert space containing the i ∈ M ) G ξ ( ξ defined by ,... , G defines a do not select a single unirrep of the phase group ξ ( | G i (instead of the M p f at the quantum level is given by the selection of the , but rather the yields the -symplectic reduction M ξ of H M 0 -symplectic reductions with 7→ ξ M H ξ H -action on ) ρ p G , In gauge theories, the bijection (4.2) implies that the quan Here, In this way, the quantization of the We can now claim that “quantization commutes with q ( : -reduction µ phase i.e. the states quotient states in result is that the ξ Quantum Indeterminacy, Gauge Symmetries and Symplectic Re not constrained to a unique value in tion in the case of the is now: plectic reduction with respect to the phase invariance is the generalization of thecases strict of G-inva given by the action of states in generalization of the bijection (4.2) is given by the follow for each unirrep obtain if we quantize the unirrep observables the action of a phase group defines a different the (non-constrained) generating functions gauge group defines a unique 0-symplectic quotient (4.1) ass of a gauge theory (which restrict the theory to the trivial un PoS(FFP14)207 , - i ) is G G p the G ) that , 0 ( ω . This 1 i . In the , p − M 0 . In other M -invariant µ iq ( G H π 2 Gabriel Catren e ows. In gauge d by quantizing obtained by su- . Therefore, it is rm i . In turn, only the gau 0 ψ | G -manifold .1), this state defines M ). This difference has G G -invariant states in unirrep of G e between a gauge theory and an le the quantum gauge symmetries he following: while the quantum ases of Heisenberg indeterminacy etries can be broken by superposing the . Now, a state ss here is that this phase invariance i able and a completely undetermined ry contains states that are , the quantization of the one-point is completely “undetermined”. In the d Hamiltonian breaks the complete indeterminacy in . According to (5.2), a translation in } king the corresponding symmetries. In with respect to non-zero values of the ial unirreps of M ) ,... auge theories to the cases in which the means that the variable acted upon by G duction G p. reps of G breaks the G-phase symmetry, p ξ i , H | ps of the gauge group. Hence, no quantum -phase invariant. Indeed, the action of of the superposition. This means that . In other terms, the introduction of an in- does not define a well-determined property ) q G is completely “undetermined”. In this way, ( i q ξ ,... in { ( ) define observables on i ξ i i | i ,... ψ p = p | ξ ξ of the momentum yields a 1-dimensional Hilbert | 8 | i M ∑ p -invariant (i.e. states transforming in the trivial unir- = whose restriction to the constraint surface can be modified without changing the state as such. i ) i gau i ψ . On the contrary, the quantization of a gauge theory only p G M | . In other terms, we can change the value of this variable | ( G i ∞ acting on of the state C ,... q q -phase invariant states carrying different unirreps (5.2) ξ Dirac observables | ∈ ) as such (i.e. modulo an overall phase factor). We can then say ξ f i , that labels the unirreps of G ,... ξ ( of the state ξ | q defined by the value i p of the momentum operator in gauge-theoretical terms as foll -phase invariance of the states M just multiplies the state by an overall phase factor of the fo i can be used to define properties of the quantum states obtaine ) i i ξ p 0 i -phase invariant states is no longer are such that the variable acted upon by , | -phase invariant quantum state p ) M ) | i G i ξ ( -invariant states, i.e. states transforming in the trivial p , containing a unique quantum state. By using the bijection (5 , ,... contains states of the form G i gau ( ξ G p | ( M G M H H According to what we have just said, the essential differenc Now, the We can also understand the fact that the position -invariant (i.e. the so-called of the state , which are in correspondence, via the bijection (4.2), with 0 gau 0 M the states determinacy in the variable i.e. the indeterminacy in the variable acted upon by G superposition of states transforming under different unir the phase invariance of quantum states encodes the extreme c principle, namely the cases givenconjugate by variable. a The well-determined important vari is point the that generalization we of wantcorresponding the to strict classical stre invariance symplectic relevant reduction inmoment takes map. g place ordinary theory endowed withstates a of Hamiltonian a group gauge action theory is are necessarily t observables on of an eigenstate the (conjugate) variable acted upon by Quantum Indeterminacy, Gauge Symmetries and Symplectic Re plays no role in the definition of contains terms, without changing the state In more usual terms, the position an important consequence regarding the possibilitythe quantum of theory brea obtained by quantizing a non-constraine are not broken by the physicalstates states, the transforming quantum in phase different symm unirreps of the phase grou state breaks the gauge symmetry. To sum up, we can say that whi not possible to superpose states carrying different unirre theories, only the observables rep of the gauge group), a non-constrained Hamiltonian theo perposing one can superpose different changes the relative phases between the terms modulo phase factors (i.e. states transforming in non-triv means that the position G case of the group of translations in symplectic quotient space a unique q PoS(FFP14)207 of , G of the q -symplectic entations i Gabriel Catren p n Modern , in M. Bitbol, P. rbit), nor that there is cannot define a property q eness or epistemic limitation, l. 74, 375-386, Springer-Verlag, ophical, and Logical . According to the conceptual d complete? -invariant. Hence, the position i p , Foundations of Physics, vol. 44, G e on the one-point bles of a gauge theory do not sin- iltonian action of the group , Princeton University Press, New M ion an observer can have about a eterminacy in the position f understanding the rationale behind den variables” capable of establish- tical interpretation of the indetermi- t because these elements are different [7], Dirac observables cannot distin- isfactory conceptual interpretation of duction mechanism which is at work at quantum phase symmetries can be e moment map) of gauge symmetries is not oretical restatement of the fact that the , in C. de Ronde, S. Aerts, and D. Aerts s neither that the theory is “incomplete” , 2nd edn., Addison-Wesley Publishing duction q , is completely undetermined. i i p | , Foundations of Physics 38, 470-487 (2008). 9 Dirac conjecture ). Consequently, the position i p , the observable } ) p , Quantization of Gauge Systems q Foundations of Mechanics ( Constituting Objectivity: Transcendental Perspectives o Geometric quantization and multiplicities of group repres { = M defined by the trivial quantization of i i p | acting on , World Scientific Publishing (2014). distinction between gauge-equivalent elements in a gauge o On classical and quantum objectivity Can classical description of physical reality be considere Quantum ontology in the light of gauge theories On the Relation Between Gauge and Phase Symmetries q (for the different values of i , The Western Ontario Series in the Philosophy of Science, vo p Probing the Meaning of Quantum Mechanics: Physical, Philos , far from defining a property of the state M should not be interpreted as a form of theoretical incomplet q i i p physical | Company, Reading (1978). Berlin (2009). Kerszberg, and J. Petitot (eds.) Physics Invent. math. 67, 515-538. (eds.) Jersey (1994). Perspectives Issue 12, 1317-1335 (2014). From a conceptual viewpoint, the fact that the Dirac observa is not a Dirac observable, i.e. it does not define an observabl [2] G. Catren, [3] G. Catren, [4] G. Catren, [7] M. Henneaux and C. Teitelboim, [1] R. Abraham and J.E. Marsden, [5] G. Catren, [6] V. Guillemin and S. Sternberg, of the quantum state guish between elements belonging to the samerepresentations gauge of orbit jus the same physicalunderstood state. as a Now, generalization the (to factpoints the th towards non-zero what values we of could th nacy characterize relations as in a quantum gauge-theore mechanics. In other terms, the ind quotients framework proposed in this article,position this is just a gauge-the Quantum Indeterminacy, Gauge Symmetries and Symplectic Re case of the ordinary (i.e. non-gauge) theory given by the Ham q translations in ing a (i.e. that it might be possible to find some hypothetical “hid gle out a particular representative in each gauge orbit mean some form of epistemicgauge restriction system. to According the to amount the of so-called informat state but rather as a consequencein of gauge the theories. same In group-theoretical this re the way, the quantum problem indeterminacy is of reduced providing to agauge the sat symmetries. (open) problem o References PoS(FFP14)207 onal Gabriel Catren , Rep. Math. Phys. , Birkhäuser, Boston s duction 10 , Graduate Studies in Mathematics, vol. 64, AMS, Reduction of symplectic manifolds with symmetry Symplectic Reduction, BRS Cohomology, and Infinite Dimensi , Annals of Physics 176, 49-113 (1987). Structure of Dynamical Systems. A Symplectic View of Physic Lectures on the Orbit Method Providence (2004). Clifford Algebras 5, 121–130 (1974). (1997). [8] A.A. Kirillov, [9] B. Kostant and S. Sternberg, Quantum Indeterminacy, Gauge Symmetries and Symplectic Re [10] J.E. Marsden and A. Weinstein, [11] J.-M. Souriau,