Isometric and metamorphic operations on the space of local fundamental measures
aa r X i v : . [ m a t h - ph ] J a n Isometric and metamorphic operations on the space of localfundamental measures
Matthias Schmidt
1, 2 Theoretische Physik II, Physikalisches Institut,Universit¨at Bayreuth, D-95440 Bayreuth, Germany H. H. Wills Physics Laboratory, University of Bristol,Royal Fort, Tyndall Avenue, Bristol BS8 1TL, United Kingdom (Dated: 15 November 2010, revised: 23 December 2010, to appear in Molecular Physics)
Abstract
We consider symmetry operations on the four-dimensional vector space that is spanned by thelocal versions of the Minkowski functionals (or fundamental measures): volume, surface, integralmean curvature, and Euler characteristic, of an underlying three-dimensional geometry. A bilinearcombination of the measures is used as a (pseudo) metric with + + −− signature, represented by a4 × . INTRODUCTION Applying classical density functional theory (DFT) requires to have an approximation forthe Helmholtz free energy as a functional of the one-body density distribution(s) [1, 2]. Forthe case of additive hard sphere mixtures, Rosenfeld’s fundamental measures theory (FMT)[3] is an approximate DFT that unified several earlier liquid state theories, including thePercus-Yevick integral equation theory and scaled-particle theory, and encapsulates theirresults in a free energy functional. Several recent reviews give a detailed account of FMTand some of its extensions and modifications [4–6]. The theory was used to address a broadvariety of interesting equilibrum phenomena, ranging from freezing to capillary behaviourof liquids. When compared to computer simulation data, theoretical results for e.g. densityprofiles and interfacial tension were typically found to be very reliable. FMT rests on buildingweighted densities via convolution with the bare density profile(s). The microscopic densityprofile ρ i ( r ) of species i gives the mean number of particles of species i in an infinitesimalvolume element at given position r and hence carries dimensions of (length) − . The weighteddensities in FMT are smoothed versions of these “real” density distributions. In Kierlik andRosinberg’s (KR) version [7] of FMT [3], there are four scalar weight functions for eachhard sphere species. Rosenfeld’s original approach that involves additional weight functionswas shown to be equivalent to the KR version [8], and was later carried much further byTarazona [9] and Cuesta et al [10]. FMT has intimate connections to methods from integralgeometry [11] via the Gauss-Bonnet theorem [12].The weight functions in FMT are quantities with dimension of negative integer powersof length, ranging from (length) − to (length) . A linear combination of pairs of weightfunctions that are convolved with each other is used to express the Mayer bond f ij ( r ), asa function of distance r . Recall that for hard sphere mixtures, the Mayer bond equals f ij ( r ) = − r < R i + R j , i.e. when the two spheres with radii R i and R i over-lap, and it vanishes otherwise. Here the subscripts i, j label the different species. Originallyproposed for hard spheres, this framework was sufficient to to derive FMTs for models suchas the Asakura-Oosawa colloid-polymer mixture [13] and the Widom-Rowlinson model [14].However, the treatment of binary non-additive hard sphere mixtures required significantmodification of the mathematical structure of FMT [15]. In particular, further weight func-tions were introduced in order to correctly model the deviation of the hard core interaction2ange between species i and j from the sum of their radii, R i + R j . The fact that thisdeviation is non-vanishing is the defining feature of non-additive hard sphere mixtures. Theadditional weight (or kernel) functions possess dimensionalities up to (length) − . They canbe grouped in a double-indexed tensorial form [15] and were shown to possess a remark-able group structure [16]. In very recent work, the FMT for non-additive hard spheres wasapplied successfully to bulk structure [17, 18] and to interfacial phenomena [18].Several features of the mathematics that underlies the FMT weight functions haveemerged [15, 16]: i) The four different position-dependent fundamental measures (in theKR formulation) can be viewed as elements of an abstract four-dimensional vector space. ii)Based on dimensional analysis, a (pseudo) metric can be defined, which can be representedby a 4 × − − signature, hence it differs both from that of Minkowskispacetime in special relativity (+ − − − ) and from that of four-dimensional Euclidian space(+ + + +). iii) Operations that are common in linear algebra, i.e. matrix multiplicationand more general contraction of tensor indices possess meaningful interpretation, see e.g.the shifting transform described in Ref. [16]. Here all product operations are carried out inFourier space and hence correspond to convolutions in real space.In the present paper we explore the mathematical structure further by focusing on sym-metry operations that leave the fundamental measure metric invariant. Our motivationcomes from the fact that careful analysis of the symmetries is central to exploiting the prop-erties of any (abstract) space. Typically, this tasks requires the identification of the linearautomorphisms that leave the metric invariant. Recall that an automorphism is a bijectivefunction that maps a space onto itself (i.e. both function value and argument are elementof the same space). Much structure can be revealed by considering infinitesimal versions, orgenerators, of the transformations. In a Lie algebra the commutator of any pair of generatorscan be represented as a linear combination of again the same generators. The coefficients ofthe linear combinations form the structure constants of the algebra.In Euclidian space, the symmetry operations that leave the metric invariant are orthogo-nal transformations, or rotations. These possess three (six) independent generators in three(four) spatial dimensions. For the case of Minkowski spacetime with three spatial and onetime-like dimension, there are three spatial rotations and three Lorentz transformations, orboosts, the latter coupling time and one of the spatial dimensions. The number of degrees3f freedom, and hence the dimensionality of the group of isometries, is independent of thesignature of the metric. However, the algebraic structure, as expressed by commutator re-lations between the respective (infinitesimal) generators of the transforms, differs for bothcases. For the case of spacetime, the resulting mathematical structure is the Lorentz group.(One refers to the Poincar´e group when four translations in the different spacetime direc-tions are added.) Here we present in detail a similar analysis for the space of Minkowskifunctionals [11]. We describe four boosts and two rotations that leave the metric invariant.These are complemented by further ten operations that change the metric and that we referto as metamorphic operations. We show that the spherical shifting operation of Ref. [16] isreadily generalized to four different types of shifting, and that the corresponding generatorscan be expressed as linear combinations of the metamorphic generators.The paper is organized as follows. In Sec. II the theory is laid out, including the de-scription of inner boosts and inner rotations as isometric transformations (Sec. II B), ofmetamorphic operations (Sec. II C), and the relationship of Jeffrey’s third-rank tensor [16]to the latter (Sec. II D). Concluding remarks are given in Sec. III.
II. TRANSFORMING THE FUNDAMENTAL MEASURESA. Metric and inner scalar product
We consider a four-dimensional real vector space with elements u = ( u , u , u , u ), where u depends on the three-dimensional argument q in Fourier space. The dependence on three-dimensional position r is then obtained by inverse Fourier transform, (2 π ) − R d q e i q · r u . Thevector components u ν , with index ν = 0 , , ,
3, are dimensional objects: u ν possesses thedimension (length) ν . Hence u is a measure of volume, u of surface, u of mean curvature,and u of Gaussian curvature. We let the u ν take on arbitrary (real) values, and hencerestrict ourselves not to cases where the measures describe an underlying geometrical body.The interpretation of the u ν in terms of geometric measures is only intended to guide theintuition, the mathematics that we present in the following is based on formal arguments.4e use the metric represented by the matrix M = , (1)hence a measure of squared “length” of a vector u is given by u t · M · u = 2( u u + u u ), wherethe superscript t indicates matrix transposition, and the dot indicates matrix multiplication.The scalar product between two vectors u and v is u t · M · v = u v + u v + u v + u v .Clearly, this is symmetric upon interchange of the vectors, i.e. u t · M · v = v t · M · u . Theeigenvalues of M are − M possesses (+ + −− )signature. As M is not positive definite (i.e. not all of its eigenvalues are positive), it canyield negative squared distances and hence constitutes not a metric in the strict sense, butone refers to a pseudo metric. While it is enirely possible to discriminate between covariantand contravariant vectors and correspondingly introduce lower and upper indices, which canbe interchanged by application of the metric, we will not do so in the following. The presentpaper is primarily concerned with second-rank tensors, and we find the (index-free) matrixnotation to be simpler, and will primarily rely on this in what follows.The hard sphere weight functions of FMT can serve as an example. These are functionsof the squared wave number q and the radius R of the hard spheres. The Fourier spaceexpressions of the KR version of the weight functions are w = c + ( qRs/ w = ( qRc + s ) / (2 q ), w = 4 πRs/q , w = 4 π ( s − qRc ) /q , where s = sin( qR ) and c = cos( qR ). Thereal space expression that corresponds, via inverse Fourier transform, to w is a unit stepfunction with range of R , i.e. Θ( R − | r | ), where Θ( · ) is the Heaviside (step) function. Withinour framework we view the w ν as the four components of a vector w . By straightforwardexplicit algebra one can show that w t · M · w yields the Fourier transform of a unit stepfunction with range 2 R , i.e. the expression for w given above, but with R replaced by2 R . Explicitly this is 2( w w + w w ) = 4 π [sin(2 qR ) − qR cos(2 qR )] /q . The significancein statistical physics stems from the fact this is (up to a trivial minus sign) the Fouriertransform of the negative Mayer function of the pair potential of hard spheres of radius R .For a mixture, the additional species possesses weight functions v ν of range R ′ , given by theabove expressions for w ν , but with R being replaced by R ′ . It is straightforward to verifythat v · M · w = 4 π [sin( q ( R + R ′ )) − q ( R + R ′ ) cos( q ( R + R ′ ))] /q , which again is the above5xpression for the unit step, w , but with R replaced by the sum of the radii, R + R ′ . Theseidentities constitute one of the central building blocks of KR’s formulation of FMT. Thegeneralization to non-additive mixtures [15] amounts to introducting 4 × w ν . This shifting operation is discussed in detailin Ref. [16]. Below in Sec. II D we give further three such “internal” shifting operations.We emphasize that all transformations that are considered here are of internal nature, i.e.act on the four-dimensional space of fundamental measures, as opposed to e.g. translationsand rotations of the underlying three-dimensional Euclidian space, which we do not considerhere.The central aim of this paper is to formulate linear automorphisms that leave the met-ric (1) invariant. We refere to such operations as isometries on the space of fundamentalmeasures. Hence one has to identify 4 × A that obey A t · M · A = A · M · A t = M , (2)which implies that a vector u and its transform A · u possess the same squared modulus.This can be seen from ( A · u ) t · M · ( A · u ) = u t · A t · M · A · u = u t · M · u , where the last equalityfollows from (2). An alternative is obtained by multiplying (2) from the right by the inverse A − , and from the left by M , and observing that M = , where is the 4 × M · A t · M = A − . (3)Note that this differs from the condition for orthogonal matrices, A t = A − . While transpo-sition can be viewed as mirroring the matrix elements on the diagonal, the operation on theleft hand side of (3) corresponds to mirroring the matrix elements on the counter diagonal. B. Inner rotations and boosts as isometries
Let us formulate the linear isometries, i.e. the automorphisms A that obey (2), by choosingappropriate generators X for each different type of transform, where X is a 4 × A are then obtained by (matrix) exponentiation. In order tosee this, consider that the expression + X dτ can be viewed as an infinitesimal transformof differential magnitude dτ . A transform by a finite amount τ can then obtained in thecontinuum limit of N -fold application of the infinitesimal transform, where each step is taken6o be of magnitude τ /N . This amounts to lim N →∞ ( + τ X /N ) N = exp( τ X ) ≡ A , wherethe result depends parametrically on τ and the form of A is specific to that of X . Herethe exponential of a matrix is defined by its power series exp( τ X ) = P ∞ m =0 ( τ m /m !) X m . Inthe following, we allow the transformation parameter τ to be dimensional, i.e. to carry anon-vanishing power of length scale.In order to allow for meaningful matrix multiplication (as is necessary for matrix expo-nentiation) the generators need to possess matrix components with suitable dimensionalities.This implies that the product of the transformation parameter and a matrix entry, τ X µν ,where µ enumerates the rows and ν enumerates the columns, with both indices runningfrom 0 to 3, must be of unit (length) µ − ν . Taking matrix powers then preserves the orderingof dimensions, i.e. the µν -component of the m -th matrix power, ( τ m X m ) µν , has the samedimensionality as τ X µν itself. Hence we can exponentiate the generators and obtain finitetransforms. Besides letting τ be a dimensional object, in the following the only furtherdependence on length scale shall be via q , the squared argument in Fourier space. Thiscorresponds to the (negative) Laplacian in the corresponding real three-dimensional space.From general arguments for four-dimensional spaces, we expect the isometry group to besix-dimensional, i.e. to possess six linearly independent generators, cf. the cases of Euclidianspace and Minkowski spacetime of special relativity mentioned above. Given a set of suchgenerators, { X α } , ennumerated by index α , a general transform is obtained as exp( P α τ α X α ),where τ α is the magnitude of the α -th transform. In principle the different contributionsto the the total transform can be disentangled via the Baker-Campbell-Hausdorff formula.This requires knowledge of the algebraic group structure, which is encoded in commutatorrelations between the different generators, as laid out below.Here we discriminate between four generators for boosts, B α , and two generators for ro-tations, D α . The subscript indicates the dimensionality; the µν -element of a given generatormatrix possesses units of (length) µ − ν − α . As laid out above, all elements along a given di-agonal possess the same dimensionality; the dimensionality then decreases (increases) byone power of length scale when moving up (down) to the next diagonal. We call booststhose generators that satisfy B α · B α = q α . Generator of rotations are those that satisfy D α · D α = − q α . The generators are not unique; one can always build linear combinations7o obtain a different formulation. Here we choose the following set of generators. B = − − , B = − q
00 0 0 q − , D = − q
00 0 0 q − , (4) B ′ = − − , B = − q − q , D = − q q − , (5)We have grouped the generators into two families, each consisting of two boosts and onerotation. The first one consists of B , B , D and is given in (4), the second one consists of B ′ , B , D and is given in (5). Both families form closed Lie algebras, constituted by thecommutator relations[ B , B ] = 2 D , [ B , D ] = 2 B , [ B , D ] = 2 q B , (6)[ B ′ , B ] = 2 D , [ B ′ , D ] = 2 B , [ B , D ] = 2 q B ′ , (7)where the commutator between two matrices X and Y is defined as [ X , Y ] = X · Y − Y · X .Members of different families commute; these are pairs of boosts: [ B , B ′ ] = [ B , B ] =[ B ′ , B ] = [ B , B ] = 0, the (only) pair of rotations: [ D , D ] = 0, and the four mixedpairs of a rotation and a boost: [ B , D ] = [ B , D ] = [ B , D ] = [ B ′ , D ] = 0. Tab. Igives an overview of the group structure in table format. All relationships can be obtainedby straightforward matrix algebra. We give a full multiplication table in Tab. II; anti-commutator relations are included for completeness. Note that in each family already thebare products (not commutators) give the result of the commutators up to a factor of 2. Asa consequence, the anti-commutators within each sub-algebra vanish, see Tab. I. Altoughcommutators between members of different sub-algebras vanish, their plain products do not,cf. Tab. II. The nine matrices that result from the products (as referred to in the off-diagonalblocks in Tab. II) will be used below in order to define further, metamorphic, operations onthe fundamental measures.It is now straightforward to calculate finite transforms via exponentiation of the respectivegenerators multiplied by its transformation parameter. Recall that the latter is a dimensional8 X , Y ] / B B D B ′ B D B D B B − D q B D − B − q B B ′ D B B − D q B ′ D − B − q B ′ X , Y ] / B ν , and rota-tions, D ν . X denotes a matrix of the leftmost column, Y one of the top row. X · Y B B D B ′ B D B D B P H F B − D q q B H F P D − B − q B − q F P ′ F ′ B ′ P H F D B B H F P ′ − D q q B ′ D F P F ′ − B − q B ′ − q { X , Y } / B B D B ′ B D B P H F B q H F P D − q F P ′ F ′ B ′ P H F B H F P ′ q D F P F ′ − q TABLE II: Left: Multiplication table X · Y for products of the generators of boosts and rotations.Right: Table of anti-commutator relationships { X , Y } / X denotes a matrix of the leftmost column, Y one of the top row. object, and that the most general finite transform is given by exp( P ν τ ν X ν ), where τ ν possessdimensions of (length) ν . Here we give only the results for the case where all parameters barone vanish. These are the following expressions for finite transformations corresponding9o (4) exp( τ B ) = e τ e τ e − τ
00 0 0 e − τ , (8)exp( τ B ) = cosh( τ q ) 0 − q sinh( τ q ) 00 cosh( τ q ) 0 q sinh( τ q ) − q − sinh( τ q ) 0 cosh( τ q ) 00 q sinh( τ q ) 0 cosh( τ q ) , (9)exp( τ D ) = cos( τ q ) 0 − q sin( τ q ) 00 cos( τ q ) 0 q sin( τ q ) q − sin( τ q ) 0 cos( τ q ) 00 − q − sin( τ q ) 0 cos( τ q ) . (10)When applied to a vector u , (8) describes a multiplication of the components u and u by e τ , and division of u and u by the same constant. Trivially, the (pseudo) squared modulus2( u u + u u ) is left unchanged. Eq. (9) is reminiscent of a hyperbolic rotation, and (10) ofan ordinary rotation. Note the difference in occurrence of the minus signs in (9) and (10).For the generators (5) we obtain the following finite transforms:exp( τ B ′ ) = e τ e − τ e τ
00 0 0 e − τ , (11)exp( τ B ) = cosh( τ q ) − q sinh( τ q ) 0 0 − q − sinh( τ q ) cosh( τ q ) 0 00 0 cosh( τ q ) q sinh( τ q )0 0 q − sinh( τ q ) cosh( τ q ) , (12)exp( τ D ) = cos( τ q ) − q sin( τ q ) 0 0 q − sin( τ q ) cos( τ q ) 0 00 0 cos( τ q ) q sin( τ q )0 0 − q − sin( τ q ) cos( τ q ) , (13)10gain (11) induces a straightforward scaling of vector components, (12) is a hyperbolicrotation and is (13) an ordinary rotation. Recall that hyperbolic rotation can be viewed asLorentz transforms (and vice versa).As a summary, we have identified six real matrices B , B ′ , B , B , D , and D , that possesthe algebraic structure shown in Tab. II. The general (real) linear group, i.e. that of all real4 × = 16 dimensional. Besides the unit matrix, this leaves nine matrices tobe considered. In the following we will use the matrices obtained as products of two isometricgenerators, cf. Tab. II. We find it interesting to investigate their action, when viewed asinfinitesimal transformations, on the space of fundamental measures. Clearly, they cannotgenerate isometries – we have exhausted these already. Hence we expect that the metricwill not be conserved under the application of these further transformations, and we willhenceforth refer to these transformations as metamorphic, as they change the underlyinggeometry in a fundamental way.The difference between automorphism and metamorphisms is reflected in the symmetryproperties of their generators. The isometric generators (4) and (5) are anti-symmetric withrespect to mirroring on the counterdiagonal, i.e. each generator X satisfies M · X t · M = − X . (14)This can be seen by inserting the infinitesimal versions A = 1 + X dτ and A − = 1 − X dτ into (3). Note that the symmetry (14) leaves 6 parameters free, which is consistent withthe dimensionality of the corresponding group of transformations (and hence the number ofgenerators). Correspondingly, metamorphic generators are symmetric under mirroring onthe counter-diagonal, i.e. they satisfy M · X t · M = X , (15)as we will see in the following. Note that the symmetry (15) leaves 10 parameters undeter-mined. C. Metamorphic transformations
We start by giving the explicit expressions for the matrices that we choose as generatorsof the metamorphic operations. As above, the index ν of a given generator X ν indicates its11imensionality. Explicit expressions for the nine different generators are as follows. F = − q − q , F = − q
00 0 0 − q , F = − q q q − , (16) H = − q − − q − , H = − q
00 0 0 − q − − , F ′ = − q − q − q − , (17) P = − − , P = − q − q q , P ′ = − q q − q . (18)Here we have grouped the nine generatores into three Abelian subgroups, given in (16),(17), and (18), respectively. Any pair of matrices from of one of these subgroups satisfies[ X µ , Y ν ] = 0. In general, the commutator between matrices from different subgroups is(up to a minus sign) a multiple of q times an isometric generator. Some of these pairscommute. The complete algebra of commutator relationships between the metamorphicgenerators is summarized in Tab. III. Remarkably, the commutator between any two pairsof these matrices either vanishes or it is a multiple of one of the isometric generators ofSec. II B. Inevitably, some of the quite compact structure of the previous subsection is lost,due to the sheer number of possible pairs. Nevertheless, note that indeed members of thesame triplet { F , F , F } , { H , H , F ′ } and { P , P , P ′ } commute with each other. We defermultiplication and anti-commutator tables to the appendix. Clearly the nine generators arenot unique. In the following section we will relate a previously obtained third-rank tensorto a linear combination of these generators. Before doing so we give explicit expressions forthe finite metamorphic transformations. 12 X , Y ] / F F F H H F ′ P P P ′ F q B ′ − q B − B − q D F q B − q B − B − q D F − q D − q D − q B ′ − q B H − q B ′ q D − D q B H − q B q D − D q B F ′ q B q B − q B − q B ′ P B B D D P q D q B ′ − q B q B P ′ q D q B − q B q B ′ B H P ′ F P F ′ F B − F ′ − P q − P ′ − F q − F − H q D − P H q − F P q − H F q B ′ H P F P ′ F F ′ B − P q − F ′ − P − F q − F − H q D − P ′ H q P q − F − H F q TABLE III: Commutator relations [ X , Y ] / X denotes a matrix of the leftmost column, Y one of the top row. The upper block of ninerows give the commutator between pairs of metamorphic generators. The lower block with 6 rowsgive the commutators betweenone isometric and one metamorphic generator. τ F ) = cos( τ q ) − q sin( τ q ) 0 0 q − sin( τ q ) cos( τ q ) 0 00 0 cos( τ q ) − q sin( τ q )0 0 q − sin( τ q ) cos( τ q ) , (19)exp( τ F ) = cos( τ q ) 0 − q sin( τ q ) 00 cos( τ q ) 0 − q sin( τ q ) q − sin( τ q ) 0 cos( τ q ) 00 q − sin( τ q ) 0 cos( τ q ) , (20)exp( τ F ) = cosh( τ q ) 0 0 − q sinh( τ q )0 cosh( τ q ) q sinh( τ q ) 00 q − sinh( τ q ) cosh( τ q ) 0 − q − sinh( τ q ) 0 0 cosh( τ q ) . (21)The second group of finite metamorphic operations isexp( τ H ) = cosh( τ q ) − q sinh( τ q ) 0 0 − q − sinh( τ q ) cosh( τ q ) 0 00 0 cosh( τ q ) − q sinh( τ q )0 0 − q − sinh( τ q ) cosh( τ q ) , (22)exp( τ H ) = cosh( τ q ) 0 − q sinh( τ q ) 00 cosh( τ q ) 0 − q sinh( τ q ) − q − sinh( τ q ) 0 cosh( τ q ) 00 − q − sinh( τ q ) 0 cosh( τ q ) , (23)exp( τ F ′ ) = cosh( τ q ) 0 0 − q sinh( τ q )0 cosh( τ q ) − q sinh( τ q ) 00 − q − sinh( τ q ) cosh( τ q ) 0 − q − sinh( τ q ) 0 0 cosh( τ q ) . (24)14nd the third group of finite metamorphic operations isexp( τ P ) = e τ e − τ e − τ
00 0 0 e τ , (25)exp( τ P ) = cos( τ q ) 0 0 − q sin( τ q )0 cos( τ q ) − q sin( τ q ) 00 q − sin( τ q ) cos( τ q ) 0 q − sin( τ q ) 0 0 cos( τ q ) , (26)exp( τ P ′ ) = cos( τ q ) 0 0 − q sin( τ q )0 cos( τ q ) q sin( τ q ) 00 − q − sin( τ q ) cos( τ q ) 0 q − sin( τ q ) 0 0 cos( τ q ) . (27) D. Morphological shifting and Jeffrey’s third-rank tensor
Based on the mathematical structure of Ref. [15], in Ref. [16] a “shifting operation” wasinvestigated that changes the radius of a sphere by a given amount R . These operations builda one-dimensional Abelian group. The generator of the group, T (referred to as ˜ G in [16]),generates the kernel K R of Ref. [15] upon exponentiation, K R = exp( RG ) ≡ exp( R T ). Thesignificance of the matrix K R lies i) in the algebraic structure: K R · K R ′ = K R ′ · K R = K R + R ′ ,and ii) in the fact that it contains the expressions for the four Kierlik-Rosinberg weightfunctions explicitly.Jeffrey’s third-rank tensor as a central object of Ref. [16] is given by the following set of15our matrices: T = , T = − q / (8 π )1 0 − q / (4 π ) 00 8 π , (28) T = − q / (64 π ) 00 − q / (4 π ) 0 − q / (64 π )1 0 − q / (4 π ) 00 1 0 0 , (29) T = − q / (8 π ) 0 − q / (32 π )0 0 − q / (64 π ) 00 0 0 − q / (8 π )1 0 0 0 , (30)where T = T · T / (8 π ), T = − q ( T ) − / (8 π ), where ( T ) − is the inverse of T .All T µ commute with each other, [ T µ , T ν ] = 0. Finite transformations are obtainedvia exponentiation as exp( χ ν T ν ), where χ ν is the transformation parameter. The fi-nite transforms commute with each other and they obey exp( P ν χ ν T ν ) exp( P ν χ ′ ν T ν ) =exp( P ν χ ′ ν T ν ) exp( P ν χ ν T ν ) = exp( P ν ( χ ν + χ ′ ν ) T ν ). [ X , Y ] / T T T T T T T T X · Y T T T T T T T T T T T π T − q π T + T − q π T T T − q π T + T − q π T − q π T − q π T T T − q π T − q π T − q π T − q π T TABLE IV: Tables for commutator relationships [ X , Y ] / X · Y for the generatorsof T µ transformations. χ T ) = e χ χ χ
00 0 0 e χ , (31)exp( χ T ) = c + qsχ / cq χ − qs ) / − q sχ / (16 π ) ( cq χ − sq ) / (16 π )( s + cqχ ) / (2 q ) c − ( qsχ ) / − (3 sq + cq χ ) / (16 π ) − q sχ / (16 π )4 πsχ /q π ( s + cqχ ) /q c − ( qsχ ) / cq χ − sq ) / π ( s − cqχ ) /q πsχ /q ( s + cqχ ) / (2 q ) c + ( qsχ ) / , (32)exp( χ T ) = g + ( gq χ ) / (8 π ) 0 − gq χ / (64 π ) 00 g − gq χ / (8 π ) 0 − gq χ / (64 π ) gχ g − ( gq χ ) / (8 π ) 00 gχ g + ( gq χ ) / (8 π ) , (33)exp( χ T ) = C − ( q Sχ ) / (16 π ) − (8 πqS + Cq χ ) / (16 π ) S/ (2 q ) − ( Cq χ ) / (16 π ) C + ( q Sχ ) / (16 π ) − qSχ / πS/q − ( Cq χ ) / πS/q + Cχ / − qSχ / q Sχ / (128 π ) − (24 πq S + Cq χ ) / (128 π )( − πqS + Cq χ ) / (128 π ) q Sχ / (128 π ) C + ( q Sχ ) / (16 π ) − (8 πqS + Cq χ ) / (16 π ) S/ (2 q ) − Cq χ / (16 π ) C − ( q Sχ ) / (16 π ) , (34)where we have used the short-hand notation s = sin( qχ ) , c = cos( qχ ), g =exp( − q χ / (8 π )), C = cos( q χ / (8 π )) , S = sin( q χ / (8 π )). Eq. (32) describes K R whensetting χ = R .It is an interesting application of the theory outlined in the previous section to try anexpress the T ν as linear combinations of the metamorphic generators. This can indeed be17one with a little algebra, yielding the result: T = , (35) T = F − H π ( P − P ′ + F − F ′ ) q + 3 P − P ′ − F + 3 F ′ πq , (36) T = q ( P − )8 π + F − H F + H π , (37) T = q ( F + H )16 π + P + P ′ − F − F ′ P + P ′ + F + 3 F ′ π . (38) III. CONCLUSIONS
In conclusions we have presented a framework for manipulating four-dimensional vectorfields u that are defined on an underlying three-dimensional Euclidian space. In real space,the relevant operations are application of the Laplace operator and building convolutions.These operations turn to multiplication by − q and the product operation in Fourier space.We have analysed the symmetries that leave the metric (1) for the four-vectors invariant.This leads to operations that either leave the metric invariant (isometric transforms) or thatchange the metric and hence the morphology that the four-vectors describe (metamorphicoperations). We have kept the nature of the four-vectors general, i.e. these can taken onaribtrary real values. This includes specific geometries (such as spheres considered in Ref.[16]), but is more general. Whether the transformations presented here help to contruct novelDFT approximations is an interesting question for future work. It would also be interestingto explore possible connections to the integral geometric framework by Hansen-Goos andMecke [19]. Acknowledgments
The published version has a dedication to a very eminent theoretical physicist in it (guesswho!).
M R Dennis, G Leithall, G Rein, and F Catanese are acknowledged for discussions,and M Burgis for a very careful reading of the manuscript. This work was supported by theEPSRC under grant EP/E065619/1 and by the SFB840/A3 of the DFG.18 ppendix A: Mixed commutator relations of isometric and metamorphic generators
Here we give further details about the algebra of commutator relations. Tab. V gives amultiplication table between metamorphic generators as well as anti-commutator relation-ships. Tab. VI gives products and anti-commutators between mixed pairs of an isometricand a morphometric generator. X · Y F F F H H F ′ P P P ′ F − q − F q F q B ′ − P ′ − q B − B − q D q H F − F − q q F − P q B − q B − B q H − q D F q F q F q − q D − q D q P F ′ − q B ′ − q B H − q B ′ − P q D q − F ′ − q H − D − q F q B H − P ′ − q B q D − F ′ q − q H − D q B − q F F ′ q B q B q P − q H − q H q F − q B − q B ′ P B B F ′ D D F P ′ P P q D q H q B ′ − q F − q B q B P ′ − q − q P P ′ q H q D q B − q B − q F q B ′ P − q P − q { X , Y } / F F F H H F ′ P P P ′ F − q − F q F − P ′ q H F − F − q q F − P q H F q F q F q q P F ′ H − P q − F ′ − q H − q F H − P ′ − F ′ q − q H − q F F ′ q P − q H − q H q F P F ′ F P ′ P P q H − q F P ′ − q − q P P ′ q H − q F P − q P − q TABLE V: Top: Multiplication table X · Y for products of the generators of metamorphic operations.Bottom: Anti-commutator relations { X , Y } / X · Y + Y · X ) / X denotes a matrix of the leftmost column, Y one of the top row. · Y F F F H H F ′ P P P ′ B D H P ′ B F P B ′ F ′ F B − F ′ − P q B q − P ′ B ′ q − F q − F D q − H q D − P − B ′ q H q − F P q − D q − H F q − B q B ′ H D P F B P ′ B F F ′ B − P q − F ′ B q B q − P − F q − F − H q D q D − B q − P ′ H q P q − F − D q − H − B q F q Y · X F F F H H F ′ P P P ′ B D − H − P ′ B − F − P B ′ − F ′ − F B F ′ P q B q P ′ B ′ q F q F D q H q D P − B ′ q − H q F − P q − D q H − F q − B q B ′ − H D − P − F B − P ′ B − F − F ′ B P q F ′ B q B q P F q F H q D q D − B q P ′ − H q − P q F − D q H − B q − F q { X , Y } / F F F H H F ′ P P P ′ B D B B ′ B B q B ′ q D q D − B ′ q − D q − B q B ′ D B B B B q B q D q D − B q − D q − B q X , and generators ofmetamorphisms, Y . Middle: Reverse product order, Y · X . Bottom: Anti-commutator relationshipsfor the same pairs.
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