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Zero-Mode, Winding Number and Quantization of Abelian Sigma

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Zero-Mode, Winding Number and Quantization of Abelian Sigma View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server DPNU-94-60 hep-th/9412174 Decemb er 1994 Zero-mo de, winding numb er and quantization of ab elian sigma mo del in (1+1) dimensions Shogo Tanimura Department of Physics, Nagoya University, Nagoya 464-01, Japan Abstract We consider the U (1) sigma mo del in the two dimensional space- 1 time S R, which is a eld-theoretical mo del p ossessing a non- trivial top ology. It is p ointed out that its top ological structure is characterized by the zero-mo de and the winding numb er. A new typ e of commutation relations is prop osed to quantize the mo del resp ecting the top ological nature. Hilb ert spaces are constructed to b e representation spaces of quantum op erators. It is shown that there are an in nite numb er of inequivalent representations as a consequence of the nontrivial top ology. The algebra generated by quantum op erators is deformed by the central extension. When the central extension is intro duced, it is shown that the zero-mo de variables and the winding variables ob ey a new commutation rela- tion, whichwe call twist relation. In addition, it is shown that the central extension makes momenta op erators ob ey anomalous com- mutators. We demonstrate that top ology enriches the structure of HEP-TH-9412174 quantum eld theories. e-mail address : [email protected] 1 Intro duction As everyone recognizes, quantum theory is established as an adequate and unique language to describ e microscopic phenomena including condensed matter physics and also particle physics. In this pap er wewould like to develop quantum theory in the direction to seek for another p ossibility which inheres in the theory itself. First we shall explain the well-known feature of quantum theory and then we shall show the direction to pursue. To formulate a quantum theory we start from commutation relations among generators. Generators de ne an algebra, which is asso ciative but not commutative. When wehave a classical theory,we ordinarily bring commutators among canonical variables by replacing the Poisson brackets. Then we construct a Hilb ert space to b e an irreducible representation space of the algebra. An element of the algebra is chosen and called a hamiltonian. Choice of a hamiltonian is restricted byphysical requirements like self-adjointness, p ositivity and symmetries. The set formed by the algebra, the representation space and the hamiltonian is called a quantum theory. Then formulation itself is nished. Next tasks are to solve it; wewanttoknow eigenvalues of various observables, esp ecially we are interested in sp ectrum of the hamiltonian, wewant to know proba- bility amplitude of transition b etween an initial state and a nal one. For physically interesting theories, those are dicult tasks. Of course, we appreciate that they are worth hard work. However it is not the direction in whichwewould like to pro ceed in this pap er. We are interested in a rather formal asp ect. A feature of quantum theory is that it requires a representation space; op erators must b e provided with op erands. A commutator in quantum theory corresp onds to aPoisson bracket in classical theory. But the Hilb ert space in quantum theory has no corresp ondence in classical theory. State vectors, the sup erp osition principle and amplitudes are characteristic concepts of quantum theory.We b egin with an algebra of op erators, then we construct a representation space. That is a unique pro cedure of quantum theory. Here arise two questions; do es a representation space exist? Is it unique? If there are inequivalent representations, calculation based on a di erent space gives a di erent answer for a physical quantity, for example, sp ectrum or amplitude. When we consider a particle in a Euclidean space, we b egin with the usual canon- ical commutation relations; [^x ;x^ ]= 0; (i; j =1;;n) (1.1) i j [^x ;p^ ]=i ; (1.2) i j ij [^p ;p^ ]=0: (1.3) i j According to von Neumann's theorem the irreducible representation of the ab ove algebra exists uniquely within a unitary equivalence class. Therefore there is no problem in choice of a Hilb ert space. Although one may use the wave function representation and another may use the harmonic oscillator representation, b oth obtain a same result for calculation of a physical quantity. Is there no need to worry ab out existence and uniqueness of a representation? Actually it is needed. Wehave encountered a situation in which the uniqueness is violated, when we consider a quantum eld theory. In a quantum eld theory we construct a representation space by de ning a vacuum state and a Fock space. It was found that in several mo dels there exist inequivalentvacuum states and they result in inequivalentFock spaces. The di erentvacua are characterized by its transformation prop erty under a certain symmetry op eration. Such a situation is called sp ontaneous symmetry breaking (SSB). The discovery of SSB op ened rich asp ects of quantum eld theories and led to deep understanding of the nature. A eld theory deals with a system which has in nite degrees of freedom, namely it is de ned with a in nite numb er of generators which are called eld variables. It is known that SSB is related to the in nity of degrees of freedom. On the other hand a particle has only nite degrees of freedom. Is there no o ccurence of inequivalent representations in a quantum theory of a particle? (Usually a quantum theory with nite degrees of freedom is called a quantum mechanics.) A strange result was found; when Ohnuki and Kitakado [1] investigated a quan- 1 tum mechanics of a particle on a circle S , they showed that there are a in nite numb er of inequivalent Hilb ert spaces. Those spaces are parametrized by a continu- ous parameter ranging from 0 to 1. What they have shown is that even a system with nite degrees of freedom can p ossess inequivalent representations when top ol- ogy of the system is nontrivial. After that work, they studied a quantum mechanics n on a sphere S (n 2) and showed existence of an in nite numb er of inequivalent Hilb ert spaces sp eci ed by a discrete index. Let us turn to eld theories. The scalar eld theory is a eld-theoretical cor- resp ondence to the quantum mechanics of a particle in a Euclidean space. This eld theory is quantized by requiring the canonical commutators and constructing the Fock space. There is also a corresp ondence in eld theories to a quantum me- chanics on a nontrivial manifold. It is a nonlinear sigma mo del b ecause it has a manifold-valued eld. For a review on nonlinear sigma mo dels, see the reference [2]. Originally the nonlinear sigma mo del is designed to describ e b ehavior of Nambu- Goldstone (NG) b osons at low energy scale. NG b osons are massless excitations asso ciated with SSB. When a continuous symmetry sp eci ed by a group G is broken to a smaller symmetry sp eci ed by a subgroup H ,vacua form a manifold whichis called a homogeneous space G=H . In this mo del NG b osons are describ ed by a eld taking values in G=H . It is already known [3] that even the quantum mechanics on G=H has inequivalent Hilb ert spaces. Therefore it is naturally exp ected that a quantum eld theory with a manifold-valued eld may p ossess inequivalent Hilb ert spaces. However in a usual approach, the nonlinear sigma mo del is quantized by the canonical quantization as the scalar eld theory and is solved by the p erturbative metho d. Thus the top ological nature of the theory is missed and only a Fock space provides a representation. If top ological prop erties of eld theories do not play an imp ortant role in physical application, we could neglect them. However we know several mo dels whichhave nontrivial top ology and in which top ology plays an imp ortant role. For instance, the sine-Gordon mo del has top ological kinks [4]; some gauge theories have top olog- ically nontrivial vacua, so-called -vacua [5]; some nonlinear sigma mo dels have the Wess-Zumino-Witten term, which re ects anomaly and top ology of vacua [6]; the con guration space of nonab elian gauge elds mo dulo gauge transformations has an extremely complicated top ology and causes the Grib ov problem [7], and so on. Hence wewould like to develop quantum eld theories resp ecting top ological nature. One of the aims of this pap er is to demonstrate that there are a lot of p ossibilities in constructing of quantum eld theories, even if they are identical as classical theo- ries. The second aim is to clarify the relation b etween top ology and quantization. In 1 the section 2 we will give a review of the quantum mechanics on S .We will discuss physical implication of the existence of inequivalent representations. The section 3 is a main part of the present pap er. There we consider a simple but nontrivial eld-theoretical mo del, the ab elian sigma mo del in (1+1) dimensions. We prop ose a de nition of an algebra, which is quite di erent from the canonical one. Then we will construct Hilb ert spaces and classify inequivalent ones.
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