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Matrix elements of the operator Tbar{T} in integrable quantum field theory

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factors. This allows the selection of finite dimensional linear subspaces of form factor solutions corresponding to operators with dimension smaller than a fixed value, and naturally suggests an identification procedure which starts from the primaries and then moves up to the descendants through discrete steps. It was argued in [15] that the absence of interaction between right- and left-moving particles at high energies in 1+1 dimensions is at the origin of a factorization of form factors when a subset of momenta becomes asymptotically large. Being non-linear in the operators the factorization equations single out specific solutions within the linear subspaces selected by the asymptotic bound and have been exploited for the identification of primary operators, in particular in the above mentioned case of the Ising model in a magnetic field [13, 14]. In this paper we investigate to which extent the ingredients discussed above are sufficient for the continuation of the programme, namely for the selection of non-trivial descendant operators. This problem, so far unexplored, is intrinsically complicated by the fact that the descendant operators enter the theory in multiplets degenerate with respect to the scaling dimension, and are classified in the conformal limit in a way which, also in consequence of operator mixing, loses much of its effectiveness in the S -matrix approach away from criticality. We concentrate our attention on the lowest scalar descendant in the operator family of the identity, i.e. the ¯, which can be defined in a massive theory through a regularized product of the operator T T components of the energy-momentum tensor [16]. In order to minimize the technicalities, we perform our analysis within the simplest interacting massive quantum field theory, namely the Lee-Yang model [17] which describes the edge singularity of the Ising model in a pure imaginary magnetic field [18, 19, 20, 21]. Our strategy is the following. We first require an upper bound on the asymptotic behavior and show that it indeed leads to a subspace of solutions with the expected dimensionality. Then we impose an asymptotic factorization condition which is ¯ up to terms which a generalization of that discussed in [15] and identifies the operator T T are subleading in the massless limit. Finally, these subleading terms are fixed exploiting the ¯ recently derived in [16]. It turns out that these conditions on the expectation values of T T conditions also determine part of the leading terms, and then provide a confirmation of the validity of the asymptotic factorization. Our construction leaves unfixed the coefficient of a single derivative operator which is known from conformal perturbation theory to represent an ¯. intrinsic additive ambiguity in the definition of the composite operator T T The paper is organized as follows. We recall the main features of the Lee-Yang model in the next section and the determination of the primary operator in the massive theory in section 3. ¯ while some final Section 4 is devoted to the identification of the solution for the operator T T considerations are collected in section 5. Three appendices contain some results and derivations which are referred to in section 4.
SISSA 48/2004/FM
¯ in integrable quantum field theory Matrix elements of the operator T T
Gesualdo Delfino and Giuliano Niccoli
International School for Advanced Studies (SISSA) via Beirut 2-4, 34014 Trieste, Italy INFN sezione di Trieste E-mail: delfino@sissa.it, niccoli@sissa.it
1
Introduction
It is commonly believed that the knowledge of the S -matrix implicitly amounts to the complete solution of a quantum field theory, in the sense that the off-shell physics (all operators and their correlation functions) can be reconstructed starting from the particle dynamics. The integrable theories in two dimensions are the only interacting quantum field theories which are exactly solved on shell, and then provide an unique framework for testing the status of this belief. The determination of the S -matrix in integrable theories follows from the combination of the general analyticity properties [1] with the elasticity and factorization of the scattering processes imposed by the existence of an infinite number of quantum integrals of motion [2]. Moving off-shell means evaluating the matrix elements (form factors) of the operators on the asymp

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matlab操作符

matlab操作符

matlab操作符from:总结了一些Matlab的操作符和特殊字符，希望对大家有用。

Operators and special characters.Arithmetic operators.plus - Plus +uplus - Unary plus +minus - Minus - uminus - Unary minus - mtimes - Matrix multiply *times - Array multiply .* mpower - Matrix power ^power - Array power .^ mldivide - Backslash or left matrix divide \ mrdivide - Slash or right matrix divide / ldivide - Left array divide .\ rdivide - Right array divide ./kron - Kronecker tensor product kron Relational operators.eq - Equal ==ne - Not equal ~=lt - Less than <gt - Greater than >le - Less than or equal <=ge - Greater than or equal >=Logical operators.relop - Short-circuit logical AND &&relop - Short-circuit logical OR ||and - Element-wise logical AND &or - Element-wise logical OR |not - Logical NOT ~xor - Logical EXCLUSIVE ORany - True if any element of vector is nonzeroall - True if all elements of vector are nonzero Special characters.colon - Colon :paren - Parentheses and subscripting ( ) paren - Brackets [ ]paren - Braces and subscripting { } punct - Function handle creation @punct - Decimal point .punct - Structure field access .punct - Parent directory ..punct - Continuation ...punct - Separator ,punct - Semicolon ;punct - Comment %punct - Invoke operating system command ! punct - Assignment =punct - Quote ' transpose - Transpose .' ctranspose - Complex conjugate transpose 'horzcat - Horizontal concatenation [,] vertcat - Vertical concatenation [;] subsasgn - Subscripted assignment ( ),{ },. subsref - Subscripted reference ( ),{ },. subsindex - Subscript indexBitwise operators.bitand - Bit-wise AND.bitcmp - Complement bits.bitor - Bit-wise OR.bitmax - Maximum floating point integer.bitxor - Bit-wise XOR.bitset - Set bit.bitget - Get bit.bitshift - Bit-wise shift.Set operators.union - Set union.unique - Set unique.intersect - Set intersection.setdiff - Set difference.setxor - Set exclusive-or.ismember - True for set member.OperatorUsage\a - Alarm (beep)\\ - Backslash\$ - Dollar sign\b - Backspace\f - Form feed\n - New line\r - Carriage return\t - Horizontal tab\v - Vertical tab\oN or \o{N} - Character of octal value NN or {N} - Character of hexadecimal value N\char - If a character has special meaning in a regular expression, precede it with backslash (\) to match it literally.。

Heavy-Quark Symmetry and Skyrmions

Heavy-Quark Symmetry and Skyrmions

Version 6.0
SNUTP-94/117 T94/149 November 1994 HEAVY-QUARK SYMMETRY AND SKYRMIONS
arXiv:hep-ph/9412302v1 15 Dec 1994
Dong-Pil Min Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-742, Korea Yongseok Oh Department of Physics, National Taiwan University, Taipei, Taiwan 10764, R.O.C. Byung-Yoon Park Department of Physics, Chungnam National University, Daejeon 305-764, Korea and Mannque Rho Service de Physique Th´ eorique, C. E. Saclay, 91191 Gif-sur-Yvette Cedex, France
ABSTRACT We review recent development on combining heavy-quark symmetry and chiral symmetry in the skyrmion structure of the baryons containing one or more heavy quarks, c (charmed) and b (bottom). We describe two approaches: One going from the chiral symmetry regime of light quarks to the heavy-quark symmetry regime which will be referred to as “bottom-up” approach and the other going down from the heavy-quark limit to the realistic finite-mass regime which will be referred to as “top-down.” A possible hidden connection between the two symmetry limits is suggested. This review is based largely on the work done – some unpublished – by the authors since several years.
高数英文版课件

高数英文版课件


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A1A 132
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(3215)5(252)33
1 0
0 1
Determinant of a Matrix
• The quantity ad – bc that appears in the rule for calculating the inverse of a 2 x 2 matrix is called the determinant of the matrix.
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– With this operation, we can solve equations that involve matrices.
• The Inverse of a Matrix
Identity Matrices
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Matrices - Operations
When the original matrix is square, transposition does not affect the elements of the main diagonal
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The identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal to their transpose since the diagonal is unaffected.
Non-commutative phase and the unitarization of GL_{p,q}(2)

Non-commutative phase and the unitarization of GL_{p,q}(2)

a r X i v :h e p -t h /0208089v 2 18 O c t 2002Non–commutative phase and the unitarization of GL p,q (2)M.Arık and B.T.Kaynak Department of Physics,Bo˘g azi¸c i University,80815Bebek,Istanbul,Turkey Abstract In this paper,imposing hermitian conjugate relations on the two–parameter deformed quantum group GL p,q (2)is studied.This results in a non-commutative phase associated with the unitarization of the quantum group.After the achievement of the quantum group U p,q (2)with pq real via a non–commutative phase,the representation of the algebra is built by means of the action of the operators constituting the U p,q (2)matrix on states.1Introduction The mathematical construction of a quantum group G q pertaining to a given Lie group G is simply a deformation of a commutative Poisson-Hopf algebra defined over G .The structure of a deformation is not only a Hopf algebra but characteristically a non-commutative algebra as well.The notion ofquantum groups in physics is widely known to be the generalization of the symmetry properties of both classical Lie groups and Lie algebras,where two different mathematical blocks,namely deformation and co–multiplication,are simultaneously imposed either on the related Lie group or on the related Lie algebra.A quantum group is defined algebraically as a quasi–triangular Hopf alge-bra.It can be either non-commutative or commutative.It is fundamentally a bi–algebra with an antipode so as to consist of either the q–deformed uni-versal enveloping algebra of the classical Lie algebra or its dual,called the matrix quantum group,which can be understood as the q–analog of a classi-cal matrix group [1].One needs four axioms,namely morphisms,in order to define a bi-algebraic structure:associativity,co–associativity,unit and co–unit.There is an additional structure,called the connecting axiom,which is needed to link the algebra to its dual one.Although the applications of quantum groups mainly concentrate on the studies of quantum integrable models using the quantum inverse scatter-ing method and non–commutative geometry,there have been many phe-nomenological applications of quantum algebras in nuclear physics,con-densed matter physics,molecular physics,quantum optics and elementary particle physics.The most important and remarkable application arose from the q–deformation of the known quantum mechanical harmonic oscillator al-gebra.The algebraic approaches to the oscillator algebras involve the known creation,annihilation and number operators.It is worth emphasizing the importance of the algebra possessing hermitian operators,giving rise to the ability of representing the physical observables.An algebra,therefore,needs to have a ∗structure to be interpreted as an algebra of observables.The simplest matrix quantum group with such a structure is SU q (2)[2].Let us now review the quantum group GL p,q (2)and then its unitary form U q,¯q (gl (2))[3].The representation matrix and the algebra of the two–parameter deformed quantum group GL p,q (2)are defined in the following way [4,5]A = a b c d,(1.1)ab =q 2ba ac =p 2ca bd =p 2db cd =q 2dcad −da =(q 2−p −2)bc bc =p 2q −2cb.(1.2)The co–multiplication ∆,the co–unit ε,and the antipode(matrix inverse)S ,whose bi–algebra is generated by the matrix elements a ,b ,c ,and d ,are given by ∆(A )= a ⊗a +b ⊗c a ⊗b +b ⊗d c ⊗a +d ⊗c c ⊗b +d ⊗d,(1.3)ε(A )= 1001,(1.4)S (A )=D −1 d −p −2b −p 2c a= d −q −2b −q 2c a D −1,(1.5)where the quantum determinant of A is defined byD ≡det p,q A =ad −q 2bc =ad −p 2cb.(1.6)The co–product and the antipode of the quantum determinant are given by ∆(D )=D ⊗D ,(1.7)S (D )=D −1.(1.8)A unitarized form of GL p,q(2),named U¯q,q(2),can be found in Jagan-nathan and Van Der Jeugt[3].It is important to notice that our notation is different from the usual one as regards the usage of the deformation parame-ters p and q.The deformation parameters p and q should be replaced by p1/2 and q1/2to obtain the usual convention in[3].The fundamental A–matrixof the quantum group is given byA= a b c d = a−¯q D c∗c D a∗ = a−qc∗Dc a∗D,(1.9)where the matrix elements satisfyac=¯q ca a D=D a ac∗=qc∗a D c∗=e2iθc∗D cc∗=c∗c,D∗D=DD∗=1aa∗+|q|2c∗c=1a∗a+c∗c=1.(1.10) Here q=|q|e iθ,p=¯q andθis a phase.The case D=1which also implies θ=0corresponds to SU q(2).In this paper an algebra obtained by imposing∗relations on the operators a,b,c and d which are the matrix elements of the quantum group GL p,q(2) will be considered.We are able to do this for pq real.In the limit p=¯q, our algebra coincides with U q,¯q(2)[3].We thus name this algebra U p,q(2). Representation of this algebra is constructed and the relationship of these representations to q–oscillators and to two–parameter coherent states are discussed.2The Unitarization of the Quantum Group GL p,q(2)with p=¯qIn order to obtain SU q(2),elements of the fundamental A–matrix,A∈GL q(2,C),are chosen in such a way that A∗=A−1=1.This choice brings about a restriction on the elements of the matrix such that b=−qc∗and d=a∗.The procedure applied to the quantum group GL p,q(2,C)to carry out the unitarization is similar to the one applied to the one–parameter deformed quantum group in order to transform GL q(2,C)into U q(2)but it is not completely the same.The most important point of the procedure we have studied is that the matrix of the quantum group should be factorized into a product which consists of the square root of the quantum determinant and a new matrix whose determinant is unity.D=δ2,(2.1)A=δA right.(2.2)The co–product and the antipode ofδare given by∆(δ)=δ⊗δ,(2.3)S(δ)=δ−1,(2.4)which are consistent with the equations(1.7)and(1.8).The next step is to impose the unitarity condition on the new matrix,resulting infinding the relation between the elements of this matrix as in the U q(2)case.Therefore, the elements of the matrix A right become the elements of SU r(2)with r∈Rr=pq=¯p¯q.(2.5)Lastly,the relations between the original matrix elements can be achieved through the relations between the new ones obtained after the unitarization of the matrix with the conditionδδ∗=s2δ∗δ,(2.6)where s is a central element of the resultant unitarized algebra of GL p,q(2). It commutes with all elements in the algebra and is also hermitian s=s∗. The co–product,co–unit and the antipode of the central element s are given by∆(s)=s⊗s,(2.7)ε(s)=1,(2.8)S(s)=s−1.(2.9)This leads to the matrix elements b and d being respectively replaced by a combination of c∗and a∗multiplied by inverse of s and a unitary operator u. The new fundamental A–matrix of the unitary quantum group is given byA= a b c d = a−¯q2s−1uc∗c s−1ua∗ .(2.10) It can be easily checked that with these relations a,c and b,d defined by(2.10) satisfy the commutation relations(1.2)of GL p,q(2).The co–product,the co–unit and the antipode of the unitary operator u are given by∆(u)=u⊗u,(2.11)ε(u)=1,(2.12)S(u)=u∗.(2.13)The whole algebra of the unitarized two–parameter quantum group,which the matrix elements obey,is given byac=p2ca a∗c∗=¯p−2c∗a∗ac∗=q2s2c∗a a∗c=¯q−2s−2ca∗aa∗−s2a∗a=(1−r2)cc∗cc∗=q¯qs2cu u∗c∗=p¯q¯p qs2c∗u u∗c=¯p qp¯q1−q2n|n−1,m ,(3.1)a∗|n,m =Here m is an integer and n is a nonnegative integer.Motivated by this,we look for a representation of the algebra(2.14)on such states.The deforma-tion parameters p and q are reparametrized in order to achieve a convenient form for the representationp= t e iθ/2q=√1−r2n te−iθ m|n−1,m−k ,(3.8) a∗|n,m =√[6]claimed that the operator d can be interpreted as a deformation of a phase operator.The work of this paper shows that a rigorous foundation for a non–commutative unitary phase operator lies in the two–parameter deformed quantum group.Whether applications such as the quantum phase operator for a quantized boson can be incorporated into this formalism will be the subject of further research.References[1]L.D.Faddeev,N.Y.Reshetikin and L.A.Takhtajan,Algebr.Anal.1(1989)178(in Russian);V.G.Drinfeld,Quantum groups,Proc.Intern.Congress of Mathematicians Vol.1(Berkeley,CA)(1986)pp.798-820;M.Jimbo,Lett.Math.Phys.11(1986)247.[2]Woronowicz S.L.,Commun.Math.Phys.111(1987)613.[3]R.Jagannathan and J.Van der Jeugt,Finite Dimensional Representa-tion of the Quantum Group GL p,q(2)using the Exponential Map from U p,q(gl(2)),J.Phys.A28(1995)2819.[4]A.Schirrmacher,J.Wess,and B.Zumino,The Two–parameter Deforma-tion of GL(2),Its Differential Calculus and Lie Algebra,Zeit.f.Physik C49(1991)317.[5]V.K.Dobrev,Duality for the Matrix Quantum Group GL p,q(2,C),J.Math.Phys.33(1992)3419.[6]M.Arık and C.Kılı¸c,Extension of the q–deformed Oscillator and Two–parameter Coherent States,J.Phys.A:Math.Gen.34(2001)7221.。