aa r X i v : . [ qu a n t - ph ] J un Quantum Blobs
Maurice A. de Gosson ∗ Universit¨at Wien,Fakult¨at f¨ur MathematikNuHAG , A-1090 Wien
June 28, 2011
Abstract
Quantum blobs are the smallest phase space units of phase spacecompatible with the uncertainty principle of quantum mechanics andhaving the symplectic group as group of symmetries. Quantum blobsare in a bijective correspondence with the squeezed coherent statesfrom standard quantum mechanics, of which they are a phase spacepicture. This allows us to propose a substitute for phase space inquantum mechanics. We study the relationship between quantumblobs with a certain class of level sets defined by Fermi for the purposeof representing geometrically quantum states.
To Basil Hiley, Physicist and Mathematician, on his 75th birthday
AMS classification 2010: 81S30; 81S10; 81Q65;
Introduction
Basil Hiley and me
It is indeed an honour and a pleasure to Contribute to Basil Hiley’s Festschrift.When I met Basil for the first time (it was in the late 90s, during my Swedish ∗ [email protected] Pastis ). Of course, I already had read a lot about the causal interpretationof quantum mechanics, but my knowledge and understanding of this theorywas merely on an abstract mathematical level. Thanks to Basil’s pedagogicalskills
Physics now entered the scene and helped me to understand some ofthe deep implications of the causal interpretation. However, Basil also wasa patient and empathetic listener, always eager to hear about new devel-opments in mathematics (Basil is not only a brilliant physicist, he also hasan excellent taste for mathematics). When I explained to him my ideas onthe uncertainty principle and introduced him to the “symplectic camel” and“quantum blobs”, he immediately became very enthusiastic and encouragedme to pursue the approach I had initiated in some recent papers. He waseven kind enough to honour me by writing a foreword to my book [10] whereI explained some of these ideas. Therefore I could do no less than to writethis modest contribution to the “Hiley Festschrift” as a tribute to my friendBasil for his Helsinki birthday party!
Contents
In this paper I establish a fundamental correspondence of a geometric naturebetween the squeezed coherent states familiar from quantum optics, andquantum blobs. The latter are related the principle of the symplectic camel,which is a deep topological property of canonical transformations, and allowa “coarse graining” of phase space in units which are symplectic deformationsof phase space balls with radius √ ~ .This paper is structured as follows. I begin by reviewing in Sect. 1 themain definitions and properties of squeezed coherent states; In Sect. 2 I in-troduce the notion of quantum blob which we discuss from a purely geometricpoint of view. In Sect. 3 the fundamental correspondence between squeezedcoherent states and quantum blobs is established; this correspondence whichis denoted by G is bijective (that is one-to-one and onto); its definition is2ade possible using the theory of the Wigner transform of Gaussian func-tions. In Sect. 4 I prove the fundamental statistical property of quantumblobs: they are a geometric picture of minimum uncertainty. Finally, inSect. 5, I shortly discuss the relationship between quantum blobs and a cer-tain level set introduced in 1930 by Enrico Fermi and which seems to havebeen almost unnoticed in the Scientific literature. The paper ends with someconjectures and a discussion of related topics I plan to develop in furtherwork. Notation
The phase space R nz ≡ R nx ⊕ R np ( n ≥
1) is equipped with the standardsymplectic form σ ( z, z ′ ) = p · x ′ − p ′ · x if z = ( x, p ), z ′ = ( x ′ , p ′ ). We arewriting x = ( x , ..., x n ), p = ( p , ..., p n ), and p · x = p x + · · · + p n x n is theusual Euclidean scalar product of p and x . Equivalently σ ( z, z ′ ) = J z · z ′ where J = (cid:18) n × n I n × n − I n × n n × n (cid:19) is the standard symplectic matrix. The group oflinear automorphisms of R nz is denoted by Sp(2 n, R ) and called the standardsymplectic group. We have S ∈ Sp(2 n, R ) if and only if S is a linear mapping R nz −→ R nz such that S T J S = J . Acknowledgement 1
This work has been supported by the Austrian Re-search Agency FWF (Projektnummer P20442-N13).
For details and complements see the seminal paper by Littlejohn [26]; Folland[8] also contains valuable information.The archetypical example is that of the fiducial (or standard, or vacuum)coherent state Φ ~ ( x ) = ( π ~ ) − n/ e −| x | / ~ (1)where the factor ( π ~ ) − n/ is introduced in order to ensure normalization. Itwas first systematically used by Schr¨odinger in 1926. The notation Φ ~ ( x ) = h x | i is also widely used in quantum mechanics. It represents the groundstate of the isotropic harmonic oscillator; alternatively it is an eigenstate ofthe annihilation operator with eigenvalue zero. More generally one wants to3onsider Gaussians of the typeΦ ~ X,Y ( x ) = ( π ~ ) − n/ (det X ) / e − ~ ( X + iY ) x · x (2)where X and Y are real symmetric n × n matrices, X positive definite; wehave || Φ ~ X,Y || L = 1. The Gaussian (2) is called a squeezed (or generalized)coherent state. Let b T ~ ( z ) be the Heisenberg–Weyl operator defined for afunction Ψ ∈ L ( R n ) by b T ~ ( z ) ψ ( x ) = e i ~ ( p · x − p · x ) ψ ( x − x ) . (3)We can still go one step further and define the shifted squeezed coherentstate Φ ~ X,Y,z ( x ) = b T ~ ( z )Φ ~ X,Y ( x ) (4)where b T ~ ( z ) is the Heisenberg–Weyl operator: if Ψ is a function on config-uration space R nx then b T ~ ( z )Ψ( x ) = e i ~ ( p · x − p · x ) Ψ( x − x ) . (5)We will write from now on M = X + iY , Φ ~ M = Φ ~ X,Y , Φ ~ M,z = b T ~ ( z )Φ ~ M . (6)The important thing is that squeezed coherent states are naturally ob-tained from the fiducial state (1) by letting metaplectic operators act on it.Let us explain this property shortly; for details see for instance de Gosson[15]. The symplectic group Sp(2 n, R ) has a covering group of order two, themetaplectic group Mp(2 n, R ). That group consists of unitary operators (themetaplectic operators) acting on L ( R n ). There are several equivalent waysto describe the metaplectic operators. For our purposes the most tractableis the following: assume that S ∈ Sp(2 n, R ) has the block-matrix form S = (cid:18) A BC D (cid:19) with det B = 0 . (7)The condition det B = 0 is not very restrictive, because one shows (de Gos-son [10, 15]) that every S ∈ Sp(2 n, R ) can be written (non uniquely however)as the product of two symplectic matrices of the type above; moreover thesymplectic matrices arising as Jacobian matrices of Hamiltonian flows deter-mined by physical Hamiltonians of the type “kinetic energy plus potential”are of this type for almost every time t . To the matrix (7) we associate thefollowing quantities: 4 A quadratic form W ( x, x ′ ) = 12 DB − x · x − B − x · x ′ + 12 B − Ax ′ · x ′ defined on the double configuration space R nx × R nx ; the matrices DB − and B − A are symmetric because S is symplectic ( W ( x, x ′ ) is oftencalled “Hamilton’s characteristic function” Goldstein [9]) in mechanics,or “eikonal” in optics; it is closely related to the notion of action deGosson [10, 15]); • The complex number ∆( W ) = i m p | det B − | where m (“Maslov in-dex”) is chosen in the following way: m = 0 or 2 if det B − > m = 1 or 3 if det B − < S are then given by b S Ψ( x ) = (cid:0) πi (cid:1) n/ ∆( W ) Z e i ~ W ( x,x ′ ) Ψ( x ′ ) d n x ′ . (8)The fact that we have two possible choices for the Maslov index shows thatthe metaplectic operators occur in pairs ± b S ; this of course is just a reflectionof the fact that Mp(2 n, R ) is a two-fold covering group of Sp(2 n, R ).The action of Mp(2 n, R ) on squeezed coherent states is given by thefollowing result: Proposition 2
Let b S ∈ Mp(2 n, R ) be one of the two metaplectic operatorscorresponding to the symplectic matrix S = (cid:18) A BC D (cid:19) (we do not make theassumption det B = 0 ). Then b S Φ ~ M = e i ~ γ ( b S ) Φ ~ M S with M S = i ( AM + iB )( CM + iD ) − (9) where e i ~ γ ( b S ) is a phase factor such that γ ( − b S ) = γ ( b S ) + iπ ~ [ the matrix CM + iD is never singular]. More generally we have: b S Φ ~ M,z = e i ~ γ ( b S ) Φ ~ M S ,Sz . (10) Proof.
See Folland [8], Littlejohn [26].This important result motivates the following definition:5 efinition 3
The set CS ( n, R ) of all squeezed coherent states consists of all { e i ~ γ Φ ~ M,z } where γ is an arbitrary real phase. We thus do not distinguish between e i ~ γ Φ ~ M,z and e i ~ γ ′ Φ ~ M,z ; we will oftenomit the prefactor e i ~ γ . Proposition 2 can now be restated in terms of a groupaction: Mp(2 n, R ) × CS ( n, R ) −→ CS ( n, R )( b S, e i ~ γ Φ ~ M ) e i ~ ( γ + γ ( b S )) Φ ~ M S . We will come back to this action in a moment and give a geometric pictureof it in terms of phase space ellipsoids.An important property of Proposition 2 above is that CS ( n, R ) is pre-served by Hamiltonian flows arising from quadratic Hamiltonian functions,i.e. Hamiltonians of the general type H ( z ) = 12 Rz · z (11)where R is a real symmetric matrix. When H is of the physical type “ki-netic energy plus potential” this amounts considering potentials which arequadratic forms Ω x · x in the position variables (generalized harmonic os-cillator): H ( z ) = 12 m | p | + 12 Ω x · x. (12)For Hamiltonians of the type (11) the flow determined by the Hamiltonequations ˙ x = ∇ p H ( x, p ) , ˙ p = −∇ x H ( x, p ) (13)consists of linear canonical transformations (Arnol’d [1], Goldstein [9], deGosson [15]). In fact, rewriting these equations in the form ˙ z = J Xz with X = − J R the explicit solution is given by z t = ( x t , p t ) = e tX z . Thematrix X belongs to the symplectic Lie algebra sp (2 n, R ) (because XJ + J X T = 0, see Folland [8] or de Gosson [10, 15]) hence the matrices S t = e tX are symplectic. For instance, for the generalized harmonic oscillator (12)the Hamilton equations are ˙ x = p/m and ˙ p = − Ω x and we have X = (cid:18) /m − Ω 0 (cid:19) .It follows from the theory of the metaplectic group that together with thetheory of covering spaces (see e.g. Folland [8], de Gosson [15]) that to the path6 S t = e tX of symplectic matrices corresponds a unique path t b S t of metaplectic operators such that b S is the identity. The remarkable fact isthat this family of operators b S t is just precisely the quantum flow determinedby Schr¨odinger’s equation i ~ ∂ Ψ ∂t = H ( x, − i ~ ∇ x )Ψ (14)where H ( x, − i ~ ∇ x ) is the (Weyl) quantization of the quadratic Hamiltonian(11); for instance when H has the physical type (12) this equation is just theusual equation i ~ ∂ Ψ ∂t = (cid:20) − ~ m ∇ x + 12 Ω x · x (cid:21) Ψ . (15)Thus the solution of (14) is given by the simple formulaΨ( x, t ) = b S t Ψ ( x ) , Ψ ( x ) = Ψ( x,
0) (16)In particular, if the initial wavefunction Ψ ( x ) is a coherent state Φ ~ M ,z Proposition 2) shows that the solution Ψ( x, t ) is explicitly given byΨ( x, t ) = e i ~ γ ( t ) Φ ~ M t ,z t ( x ) (17) • z t = ( x t , p t ) is the solution of Hamilton’s equations ˙ x = ∇ x H , ˙ p = −∇ p H passing through the point z at time t = 0; • M t is calculated using formula (9): write S t as a symplectic blockmatrix (cid:18) A t B t C t D t (cid:19) ; then M t = i ( A t M + iB t )( C t M + iD t ) − ; (18)One proves (see for instance Nazaikiinskii et al. [27]) that • The phase γ ( t ), is the symmetrized action integral γ ( t ) = Z t (cid:0) σ ( z τ , ˙ z τ ) − H (cid:1) dτ. (19)7 Quantum Blobs
Quantum blobs are minimum uncertainty units which are measured using notvolume, but rather symplectic capacity, which has the properties of an area–that is of action! Besides the fact that they allow a geometric descriptionof the uncertainty principle [11, 12, 13, 14, 15] (of which the reader will finda precise description in next subsection), we are going to see that they areintimately related to the notion of squeezed coherent states, of which it canbe considered as a phase space geometric picture.By definition, a quantum blob is a subset QB n = QB n ( z , S ) of R nz which can be deformed into the phase space ball B n ( √ ~ ) : | z | ≤ ~ usingonly translations and linear canonical transformations S ∈ Sp(2 n, R ). Equiv-alently, QB n is an ellipsoid obtained from B n ( √ ~ ) by an affine symplectictransformation. More precisely: Definition 4
Let S ∈ Sp(2 n, R ) and z ∈ R nz . Then QB n ( z , S ) = T ( z ) SB n ( √ ~ ) where T ( z ) is the translation operator z z + z . Equivalently, it is theset QB n ( z , S ) = { z : ( S − ) T S − ( z − z ) ≤ ~ } where we are writing ( S − ) T S − ( z − z ) for ( S − ) T S − ( z − z ) . The set ofall quantum blobs in phase space R nz is denoted QB (2 n, R ) . One shows (de Gosson [15], de Gosson and Luef [19]) that a quantumblob QB n ( z , S ) is characterized by the two following equivalent properties: • The intersection of the ellipsoid QB n ( z , S ) with a plane passing through z and parallel to any of the plane of canonically conjugate coordinates x j , p j in R nz is an ellipse with area π ( √ ~ ) = h ; that area is called the symplectic capacity of the quantum blob QB n ( z , S ) (we will discussmore in detail this notion in a moment); • The supremum of the set of all numbers πR such that the ball B n ( √ R ) : | z | ≤ R can be embedded into QB n ( z , S ) using canonical transforma-tions (linear, or not) is π ( √ ~ ) . Hence no phase space ball with radius R > √ ~ can be “squeezed” inside QB n ( z , S ) using only canonicaltransformations (Gromov’s non-squeezing theorem [20], alias the prin-ciple of the symplectic camel).
8t turns out (de Gosson [15]) that in the first of these conditions one canreplace the plane of conjugate coordinates with any symplectic plane (a sym-plectic plane is a two-dimensional subspace of R nz on which the restrictionof the symplectic form σ is again a symplectic form).Clearly there is a natural actionSp(2 n, R ) × QB (2 n, R ) −→ QB (2 n, R )( S, QB n ( z , S )) S [ QB n ( z , S )]of symplectic matrices on quantum blobs: for S ′ ∈ Sp(2 n, R ) we have S ′ T ( z ) = T ( S ′− z ) S ′ and hence S ′ [ QB n ( z , S )] = T ( S ′− z ) S ′ SB n ( √ ~ ) = QB n ( z , S ′ S ) . (20)Conversely: Proposition 5
Let G ∈ Sp(2 n, R ) be positive-definite and symmetric. Theset { z : G ( z − z ) ≤ ~ } is a quantum blob QB n ( z , S ) . Proof.
As a consequence of the symplectic polar decomposition theorem (seee.g. de Gosson [15]) there exists S ∈ Sp(2 n, R ) such that G = ( S − ) T S − hence the condition G ( z − z ) ≤ ~ is equivalent to ( S − ) T S − ( z − z ) ≤ ~ .The symplectic matrix S defining a given quantum blob is not unique;one shows (see de Gosson [15]) that QB n ( z , S ) = QB n ( z , S ′ ) if and onlyif S ′ = SU where U is a symplectic rotation, i.e. an element of the subgroup U ( n ) = Sp(2 n, R ) ∩ O (2 n, R ) of the symplectic group. This property reflectsthe invariance of phase space balls centered at the origin under rotations. Aconsequence of this fact is that we have the following topological identification(de Gosson [14]): QB (2 n, R ) ≡ R n ( n +1) × R n ≡ R n ( n +3) . Thus, if we view QB (2 n, R ) as a “quantum phase space” its topologicaldimension n ( n + 3) is much larger than that, 2 n , of the classical phase space,even when n = 1 (in the latter case dim QB (2 , R ) = 3, which is easilyunderstood as follows: one need one parameter to specify the centre of thequantum blob (which is here an ellipse with area h/ x -axis. A similar interpretation applies in higher dimensions.9et us briefly compare quantum blobs to the usual quantum cells fromstatistical mechanics. A quantum cell is typically a phase space cube withvolume ( √ h ) n = h . The first obvious remark is that these cells do nothave any symmetry under general symplectic transformations; while such atransformation preserves volume, a cube will in general be distorted into amultidimensional polyhedron. But what is more striking is the comparisonof volumes. Since a quantum blob is obtained from the ball B n ( √ ~ ) by avolume-preserving transformation its volume is given byVol (cid:0) QB n ( z , S ) (cid:1) = h n n !2 n and is hence n !2 n smaller than that of a quantum cell. For instance, inthe case of the physical three-dimensional configuration space this leads toa factor of 48. In the case of a macroscopic system with n = 10 this factbecomes unimaginably large. This is in strong contrast with the fact thatthe orthogonal projection of a quantum blob on any plane x j , p j of conjugatecoordinates (or, more generally, on any symplectic plane) is an ellipse witharea equal to π ~ = h/ G Recall that the Wigner transform of a pure state Ψ is given by W Ψ( z ) = (cid:0) π ~ (cid:1) n Z e − i ~ p · y Ψ( x + y )Ψ ∗ ( x − y ) d n y (21)where the star ∗ denotes complex conjugation.The Wigner transform of the fiducial coherent state Φ ~ is given by W Φ ~ ( z ) = ( π ~ ) − n e − ~ | z | . . More generally [15, 26] the Wigner transform W Φ ~ M ( z ) = (cid:0) π ~ (cid:1) n Z e − i ~ p · y Φ ~ M ( x + y )Φ ~ M ( x − y ) ∗ d n y (22)of the squeezed coherent state Φ ~ M = Φ ~ X,Y is given by the formula: W Φ ~ M ( z ) = ( π ~ ) − n e − ~ Gz · z (23)10here G is the real 2 n × n matrix G = (cid:18) X + Y X − Y Y X − X − Y X − (cid:19) . (24)Notice that G does not contain the parameter ~ . It turns out that G is bothpositive definite and symplectic; in fact G = S T S where S = (cid:18) X / X − / Y X − / (cid:19) ∈ Sp(2 n, R ) . (25)The same analysis applies to Φ ~ M,z ( z ). Letting the translation operator T ( z ) : z z + z act on functions on phase space by the rule T ( z ) f ( z ) = f ( z − z ) and its quantum variant, the Heisenberg–Weyl operator (26) wehave the translational property W ( b T ~ ( z ) ψ )( z ) = T ( z ) W ( ψ )( z ) (26)and hence, in particular W Φ ~ M,z ( z ) = ( π ~ ) − n e − ~ G ( z − z ) . (27)Let us now state and prove the following essential correspondence resultwhich identifies squeezed coherent states with quantum blobs: Proposition 6
There is a bijective correspondence G : CS ( n, R ) ←→ QB (2 n, R ) between coherent states and quantum blobs. That correspondence is definedas follows: if W Φ ~ M,z ( z ) = ( π ~ ) − n e − ~ G ( z − z ) then we have G [Φ ~ M,z ] = { z : G ( z − z ) ≤ ~ } = QB n ( z , S − ) (28) where the symplectic matrix S is given by formula (25) above. roof. While the definition of the correspondence G is straightforward, it isnot immediately clear why it should be bijective. Let us first show that it isone-to-one. Suppose that G [Φ ~ M,z ] = G [Φ ~ M ′ ,z ′ ], that is { z : G ( z − z ) ≤ ~ } = { z : G ′ ( z − z ′ ) ≤ ~ } . We must then have G = G ′ and z = z ′ so that W Φ ~ M,z ( z ) = W Φ ~ M ′ ,z ′ ( z );since the Wigner transform of a function Ψ determines uniquely determinesΨ up to a unimodular factor we have Φ ~ M ′ ,z ′ = e i ~ γ Φ ~ M,z for some real phase γ .Let us next show that G is onto; this will at the same time yield a procedurefor calculating the inverse of G . Assume that QB n (0 , S − ) = S − B n ( √ ~ )is a quantum blob centered at the origin. One can factorize the matrix S − as follows (“pre-Iwasawa factorization”; cf. [15], § S − = (cid:18) L Q L − (cid:19) (cid:18) A − BB A (cid:19) where the symmetric matrix L is given by L = ( D T D + B T B ) / (29)is symmetric positive definite, Q = − ( C T D + A T B )( D T D + B T B ) − / (30)with A + iB ∈ U ( n, C ). The matrix (cid:18) A − BB A (cid:19) is thus a symplectic rotationand, as such, leaves any ball centered at the origin invariant. Setting X / = L and Y = X / Q it follows that we have S − h B n ( √ ~ ) i = (cid:18) X / X − / Y X − / (cid:19) B n ( √ ~ );the quantum blob QB n (0 , S − ) is thus represented by Gz · z ≤ ~ where G = S T S is of the type (24); define now Φ ~ M = Φ ~ X,Y by assigning to X and Y the values L and X / Q found above. The argument generalizes ina straightforward way to quantum blobs with arbitrary centre.In view of the correspondence between squeezed coherent states and quan-tum blobs, we can give a phase space picture of formula (17) for the timeevolution of a squeezed coherent state when the Hamiltonian function isquadratic. Let us study this deformation in some detail.We claim that an initial quantum blob becomes after time t a new quan-tum blob which is just its image by the classical flow S t :12 roposition 7 After time t the initial quantum blob QB n ( z , S ) becomesthe quantum blob S t [ QB n ( z , S )] = QB n ( z t , S S t ) . Thus, the quantum motion of coherent states induces the classical motion forthe corresponding quantum blob.
Proof.
At initial time we are in presence of an initial quantum blob QB n ( z , S ),set of all phase space points z such that G ( z − z )( z − z ) ≤ ~ with G =( S − ) T S − . Let us calculate the Wigner transform W Ψ( z, t ) = (cid:0) π ~ (cid:1) n Z e − i ~ p · y Ψ( x + y, t )Ψ ∗ ( x − y, t ) d n y (31)of the solution Ψ( z, t ) of Schr¨odinger’s equation (14). Using formula (16))together with the symplectic covariance of the Wigner transform (de Gosson[15]) we have W Ψ( z, t ) = W ( b S t Φ ~ M ,z )( z ) = W Φ ~ M ,z ( S − t z ) . that is, in view of formula (23) giving the Wigner transform of Φ ~ M ,z : W Ψ( z, t ) = ( π ~ ) − n exp (cid:20) − ~ ( S − t ) T G ( S − t z − z ) (cid:21) . (32)= ( π ~ ) − n exp (cid:20) − ~ ( S − t ) T G S − t ( z − z t ) (cid:21) . (33)It follows that the initial quantum blob has become the ellipsoid defined by( S − t ) T G S − t ( z − z t ) ≤ ~ which proves our claim. G We begin by recalling the notion of symplectic capacity, which was alreadymentioned briefly in the beginning of this paper after the definition of quan-tum blobs. See Hofer–Zehnder [25], Polterovich [29], or de Gosson [16] and13e Gosson and Luef [19] for a review of this notion from point of view easilyaccessible to physicists.A symplectic capacity on phase space R nz assigns to every subset Ω of R nz a number c (Ω) ≥
0, or + ∞ . This assignment must obey the followingrules: (SC1) If Ω ⊂ Ω ′ then c (Ω) ≤ c (Ω ′ ); (SC2) If f is a canonical transformation then c ( f (Ω)) = c (Ω); (SC3) If λ is a real number then c ( λ Ω) = λ c (Ω); here λ Ω is the set of allpoints λz when z ∈ Ω; (SC4) We have c ( B n ( R )) = πR = c ( Z nj ( R )); here B n ( R ) is the ball | x | + | p | ≤ R and Z nj ( R ) the cylinder x j + p j ≤ R .There exist infinitely many symplectic capacities, however the construc-tion of any of them is notoriously difficult (the fact that symplectic capacitiesexist is actually equivalent to Gromov’s non-squeezing theorem [20]). How-ever they all agree on phase space ellipsoids. In fact: Proposition 8
Let W : M z · z ≤ ~ where M is a symmetric positive definite n × n matrix. We have c ( W ) = π ~ /λ max (34) for every symplectic capacity c ; here λ max is the largest symplectic eigenvalueof M. The proof of this result is based on a symplectic diagonalisation of M ;see de Gosson [15], Hofer–Zehnder [25], Polterovich [29], and the referencestherein. Recall that the symplectic eigenvalues of M are defined as follows:the eigenvalues of the matrix J M are of the type ± iλ j with λ j >
0; thesequence ( λ , ..., λ n ) is then the symplectic spectrum of M and the λ j thesymplectic eigenvalues.The smallest symplectic capacity is denoted by c min (“Gromov width”):by definition c min (Ω) is the supremum of all numbers πR such that thereexists a canonical transformation such that f ( B n ( R )) ⊂ Ω. The fact that c min really is a symplectic capacity follows from Gromov’s [20] symplectic non-squeezing theorem. For a discussion of Gromov’s theorem (and comments)14rom the physicist’s point of view see de Gosson [16], de Gosson and Luef[19].Let now K be an arbitrary real symmetric positive-definite matrix oforder 2 n and define the normalized phase space Gaussian W K ( z ) = ( π ~ ) − n/ (det K ) / e − ~ Kz · z . When K = G ∈ Sp(2 n, R ) the Gaussian W K ( z ) is the Wigner transform ofsome squeezed coherent state. Following Littlejohn [26] we define a matrixΣ by the relation Σ = ~ K − (35)hence W K ( z ) takes the familiar form W K ( z ) = (2 π ) − n (det Σ) − / e − Σ − z · z suggesting to interpret Σ as the covariance matrix of a normal probabilitydistribution centered at the origin. We will write Σ in block formΣ = (cid:18) ∆( X, X ) ∆(
X, P )∆(
P, X ) ∆(
P, P ) (cid:19) where each block has dimension n × n and ∆( P, X ) = ∆(
X, P ) T ; we use thenotation ∆( X, X ) = (Cov( x j , x k )) ≤ j,k ≤ n ∆( X, P ) = (Cov( x j , p k )) ≤ j,k ≤ n ∆( P, P ) = (Cov( p j , p k )) ≤ j,k ≤ n and set (∆ x j ) = Cov( x j , x j ) , (∆ p j ) = Cov( p j , p j )for 1 ≤ j ≤ n . The essential observation we make is: Proposition 9
Consider the phase space ellipsoid W : Σ − z · z ≤ . Thetopological condition c ( W ) ≥ ~ (36) implies the Robertson–Schr¨odinger inequalities (∆ x j ) (∆ p j ) ≥ Cov( x j , p j ) + 14 ~ (37) for ≤ j ≤ n hence, in particular, the Heisenberg uncertainty relations ∆ x j ∆ p j ≥ ~ . i ~ J is Hermitian positive semi-definite (38)implies the Robertson–Schr¨odinger inequalities (37) (but it is not equivalentto it: see de Gosson [16] for a counterexample). Some algebra together witha formula giving the symplectic capacity of an ellipsoid, then shows thatconditions (38) and (36) are equivalent. Notice that the matrix Σ + i ~ J isalways Hermitian since (Σ+ i ~ J ) ∗ = Σ − i ~ J T and J T = − J . We mention thatsymplectic capacities can be used as well for the study of the more generaluncertainty principle related to non-commutative quantum mechanics as wehave shown in de Gosson [17].Suppose now that the covariances defined above correspond to some quan-tum state Ψ (pure or mixed). The Robertson–Schr¨odinger inequalities (37)are saturated (i.e. they become equalities) exactly when that state is asqueezed coherent state Φ ~ M where M = X + iY is determined via the Wignertransform of Φ ~ M (cf. (35)) W Φ ~ M ( z ) = ( π ~ ) − n e − ~ Gz · z , G = ~ − . For instance if Φ ~ M is the fiducial coherent state Φ ~ all the covariances vanishand the inequalities (37) reduce to ∆ x j ∆ p j = ~ for 1 ≤ j ≤ n . g FIn a largely forgotten paper from 1930 Fermi [7] associates to every quantumstate Ψ a certain hypersurface g F ( x, p ) = 0. Fermi’s paper has recently beenrediscovered by Benenti [2] and Benenti and Strini [3]; in particular theseauthors give a heuristic comparison of the function g F and the Wigner trans-form W Ψ. Let us shortly study the relationship between Fermi’s functionand the notion of quantum blob. The starting point is Fermi’s observationthat the state of a quantum system may be defined in two different (butequivalent) ways, namely by its wavefunction Ψ or by measuring a certainphysical quantity whose definition goes as follows. Writing the wavefunctionin polar form Ψ( x ) = R ( x ) e i Φ( x ) / ℏ ( R ( x ) ≥ x ) real) one verifies by16 straightforward calculation that Ψ is a solution of the partial differentialequation b g F Ψ = 0 (39)where b g F = ( − i ~ ∇ x − ∇ x Φ) + ~ ∇ x RR . (40)The equation (39) seems at first sight to be ad hoc and somewhat mysterious.However much of the mystery disappears if one remarks that this equationis obtained by the gauge transform p −→ p − ∇ x Φ from the trivial equation (cid:18) − ~ ∇ x + ~ ∇ x RR (cid:19) R = 0 . (41)Consider now the Weyl symbol of the operator b g F ; it is the real function g F ( x, p ) = ( p − ∇ x Φ) + ~ ∇ x RR . (42)When ∇ x R/R < g F ( x, p ) = 0 determines a hypersurface H F in phase space R nz which Fermi ultimately identifies with the state Ψ. Letus examine the relation between Fermi’s Ansatz and the notion of quantumblob we have introduced in this paper. Let Φ ~ M = Φ ~ X,Y be the squeezedcoherent state defined by Eqn. (2); we have in this case Φ( x ) = − Y x · x and R ( x ) = e − Xx · x/ ~ hence Fermi’s function is g F ( x, p ) = ( p + Y x ) + X x · x − ~ Tr X (43)where Tr X is the trace of the matrix X (note that Tr X > X ispositive definite). The hypersurface H F is thus the closed hypersurface M F z · z = ~ with M F = 1Tr X (cid:18) X + Y YY I (cid:19) . (44)Recall now that the Wigner transform of Φ ~ M is the function W Φ ~ M ( z ) =( π ~ ) − n e − ~ Gz · z where (formulas (24) and (25)) G = (cid:18) X + Y X − Y Y X − X − Y X − (cid:19) = S T S (45)and S is the symplectic matrix S = (cid:18) X / X − / Y X − / (cid:19) . (46)17n immediate calculation shows that the matrices M F and G are related bythe formula M F = 1Tr X S T (cid:18) X X (cid:19) S. (47)Let us consider the “Fermi ellipsoid” W F : M F z · z ≤ ~ bounded by thehypersurface H F . Proposition 10 (i) There exist symplectic coordinates in which the Fermiellipsoid W F : M F z · z ≤ ~ is represented by the inequality Xx · x + Xp · p ≤ ~ Tr X (48) or by the inequality N X j =1 λ j ( x j + p j ) ≤ ~ Tr X (49) where λ , ..., λ n are the eigenvalues of X ;(ii) We have c ( W F ) = π Tr Xλ max ~ ≥ h (50) where λ max is the largest eigenvalue of M F and h ≤ c ( W F ) ≤ nh . (51) Proof. (i) In view of (47) the inequality M F z · z ≤ ~ is equivalent to (cid:18) X X (cid:19) u · u ≤ ~ Tr X with u = Sz . Let U be a rotation in R n diagonalising X , that is X = U T DU with D = diag( λ , ..., λ n ). Setting v = (cid:18) U U (cid:19) u the inequality M F z · z ≤ ~ is now equivalent to (49) and one concludes bynoting that the matrix R = (cid:18) U U (cid:19) is in U ( n ) (i.e. a symplectic rotation).(ii) Since symplectic capacities are invariant by symplectic transformations,it suffices to prove formula (50) when W F is given by Eqn. (48) or by Eqn.(49). In view of Proposition 8 we have c ( W F ) = π ~ /λ max and the equality in(50) follows noting that the symplectic spectrum of X consists of precisely theeigenvalues of X . The inequality c ( W F ) ≥ h is obvious since Tr X ≥ λ max c ( W F ) ≤ nh/ X ≤ nλ max .In view of the double inequality (51) Fermi ellipsoids are not in generalquantum blobs (except for n = 1). However each of these ellipsoids containsa quantum blob. To see this it suffices to show that the ellipsoid defined by(49) contains the ball B ( √ ~ ) (because the image of a quantum blob by alinear symplectic transformation is again a quantum blob). Now, if ( x, p ) isin B ( √ ~ ) then N X j =1 λ j Tr X ( x j + p j ) ≤ N X j =1 ( x j + p j ) ≤ ~ since λ j / Tr X ≤ ~ ( x ) = ( π ~ ) − n/ e −| x | / ~ we have X = I and Y = 0 hence theFermi ellipsoid W F is the disk | x | + | p | ≤ n ~ whose symplectic capacity is nπ ~ = nh/
2. The operator (40) is here b g F = − ~ ∇ x + | x | − n ~ (52)and the relation b g F Φ ~ = 0 is hence equivalent to ( − ~ ∇ x + | x | )Φ ~ = n ~ Φ ~ (53)which simply states the well–known fact that Φ ~ is an eigenvector of theharmonic oscillator Hamiltonian b H = ( − ~ ∇ x + | x | ) corresponding to thefirst energy level E = n ~ . One easily verifies that if Ψ ~ is the tensorproduct of n copies of the (unnormalised) Hermite functions xe − x / ~ thenthe equation b g F Ψ ~ = 0 is equivalent to ( − ~ ∇ x + | x | )Ψ ~ = n ~ Ψ ~ . (54)The argument may be repeated, and one finds that the Fermi equation (39)corresponding to a Hermite function, is always equivalent to the eigenstateequation for the harmonic oscillator corresponding to that function.The discussion above can be generalised, using metaplectic covarianceproperties, to the case of quantum states of operators corresponding to ar-bitrary Hamiltonians H = M z · z where M is symmetric positive definite(generalised harmonic oscillator). It is certainly worthwhile studying whathappens in more general cases where the quantum states are no longer Gaus-sians; see the following discussion. 19 Concluding Remarks and Perspectives
Using the correspondence G defined in Section 3 we have sees that quan-tum blobs exactly correspond to those quantum states which have minimumuncertainty in the sense of Robertson–Schr¨odinger. This justifies our claimthat quantum blobs represent the smallest regions of phase space which makesense from a quantum-mechanical perspective. In fact, contrarily to what isoften believed the Heisenberg inequalities and their stronger version, theRobertson–Schr¨odinger inequalities (37), are not a statement about the ac-curacy of our measurement instruments; their derivation assumes on thecontrary perfect instruments . The correct interpretation of these inequalitiesis the following (see e.g. Peres [28], p.93): if the same preparation procedureis repeated a large number of times, and is followed by either by a measure-ment of x j , or by a measurement of p j , the results obtained have standarddeviations ∆ x j and ∆ p j satisfying these inequalities. Such a process thusmakes clear the impossibility of talking about points in phase space havingsome intrinsic meaning (cf. Butterfield’s paper [5] refuting “pointillisme”).We note that in [6] Dragoman uses the partition of phase space in quantumblobs to propose a new formulation of quantum mechanics, based on thefollowing postulates: Axiom 11
It is not possible to localize a quantum particle in a phase spaceregions smaller that a quantum blob;
Axiom 12
The phase space extent of a quantum particle is smaller than aquantum blob.
These postulates and their implications for quantum physics certainlydeserve to be discussed further.In a recent paper [18] Hiley and I study a version of the quantum Zenoparadox for the Bohm trajectory of a sharply located particle modelled by aDirac distribution. We showed in this paper that such a recorded quantumtrajectory (in, for instance, a bubble chamber) is just the classical trajectorypredicted by standard Hamiltonian mechanics. It would be both very inter-esting and realistic to study this kind of quantum Zeno effect by replacingthe point-like particle by a squeezed coherent state, that is, equivalently, bya quantum blob. A good starting point could be Hiley [21] where the re-lationship between the Wigner–Moyal and Bohm approaches is elucidated;20lso the connections with the ideas of Hiley and collaborators in [22, 23, 24]could be useful here. We have seen in Proposition 7 that a quantum blobevolves classically under the action of the linear Hamiltonian flow deter-mined by a quadratic Hamiltonian. Of course quadratic Hamiltonians areof a very particular type; the result above remains approximately valid forarbitrary physical Hamiltonians, and this with an excellent approximationduring generically very large times (Ehrenfest time, as it is called in the the-ory of quantum revivals). This observation could allow us to prove, usingthe correspondence G , the following conjecture considerably extending theresults in de Gosson and Hiley [18]: Conjecture 13
When we continuously observe the motion of a quantumblob we see its classical Hamiltonian motion; i.e. an initial quantum blob QB n will be transformed in the set f Ht ( QB n ) after time t ; here f Ht is theclassical Hamilton flow (Arnol’d [1], Goldstein [9]). In Section 5 we briefly discussed some elementary properties of the Fermifunction g ΨF and of the associated Fermi ellipsoid W F . The discussion wasactually limited to Gaussian states. We make the following conjecture: Conjecture 14
Let Ψ be a quantum state for which the Fermi equation g F ( x, p ) defines a hypersurface in phase space bounding a compact set Ω F .Then there exists a symplectic capacity c such that c (Ω F ) ≥ h and Ω F con-tains a quantum blob. The observant Reader will perhaps have noticed that the equation g F ( x, p ) =0 for a system of particles with mass m can be rewritten12 m ( p − ∇ x Φ) + Q = 0if one introduces the quantum potential Q = − ~ m ∇ x RR familiar from the Bohmian approach top quantum mechanics (see Bohm andHiley [4]). There thus seems to be a deep connection between this theoryand the phase space approach which certainly deserves to be elucidated andextended. 21 am sure that Basil will be excited by these possibilities, and I lookforward writing new papers with him about the truly fascinating topic ofquantum phase space! Happy birthday, Basil! References [1] Arnold, V.I.: Mathematical Methods of Classical Mechanics, GraduateTexts in Mathematics, 2nd edition, Springer-Verlag (1989)[2] Benenti, G.: Gaussian wave packets in phase space: The Fermi g F func-tion, Am. J. Phys. (6), 546–551 (2009)[3] Benenti, G., Strini, G.: Quantum mechanics in phase space: first ordercomparison between the Wigner and the Fermi function, Eur. Phys. J.D 57, 117–121 (2010)[4] Bohm, D., Hiley, B.: The Undivided Universe: An Ontological Interpre-tation of Quantum Theory. London & New York: Routledge (1993)[5] Butterfield, J.: Against Pointillisme about Mechanics, Br. J. Philos. Sci. (4), 709–753 (2006) DOI: 10.1093/bjps/axl026[6] Dragoman, D.: Phase Space Formulation of Quantum Mechanics. In-sight into the Measurement Problem, Phys. Scr. , 290–296 (2005)[7] Fermi, E.: Rend. Lincei , 980 (1930); reprinted in Nuovo Cimento ,361 (1930)[8] Folland, G.B.: Harmonic Analysis in Phase space, Annals of Mathemat-ics studies, Princeton University Press, Princeton, N.J. (1981)[9] Goldstein, H.: Classical Mechanics. Addison–Wesley, (1950), 2nd edi-tion, (1980), 3d edition (2002)[10] de Gosson, M.: The Principles of Newtonian and Quantum Mechanics:The need for Planck’s constant, h . With a foreword by Basil Hiley.Imperial College Press (2001) 2211] de Gosson, M.: The “symplectic camel principle” and semiclassical me-chanics. J. Phys. A: Math. Gen. (32), 6825–6851 (2002)[12] de Gosson, M.: Phase Space Quantization and the Uncertainty Princi-ple. Phys. Lett. A, /5-6 365–369 (2003)[13] de Gosson, M.: The optimal pure Gaussian state canonically associatedto a Gaussian quantum state. Phys. Lett. A, :3–4, 161–167 (2004)[14] de Gosson, M.: Cellules quantiques symplectiques et fonctions deHusimi–Wigner. Bull. Sci. Math. (2006)[16] de Gosson, M.: The Symplectic Camel and the Uncertainty Principle:The Tip of an Iceberg? Found. Phys. , 131–179 (2009), DOI10.1016/j.physrep.2009.08.001[20] Gromov, M.: Pseudoholomorphic curves in symplectic manifolds, In-vent. Math., , 307–347 (1985)[21] Hiley, B.J.: On the Relationship between the Wigner-Moyal and BohmApproaches to Quantum Mechanics: A Step to a More General Theory?Found. Phys. , 356–367 (2009)2322] Hiley, B.J.: Non-Commutative Geometry, the Bohm Interpretation andthe Mind-Matter Relationship. In Proc. CASYS’2000, Li`ege, Belgium,Aug. 7–12, 2000.[23] Hiley, B.J. Callaghan, R.E.: Delayed-choice experiments and the Bohmapproach. Phys. Scr.138(