DOCSLIB.ORG

4 BRST Quantization of the Bosonic String

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
4 BRST Quantization of the Bosonic String 4 BRST quantization of the bosonic string 4.1 BRST quantization The BRST formalism is a consistent quantization prescription for a gauge theory at the level of path integral, assuming that the gauge algebra is not field dependent. To begin with, consider an action S[φ], where φ collectively denotes all fields in the theory, that is invariant with respect to gauge symmetries of the form δφ = dµ ✏↵δ φ ✏↵δ φ. (4.1) ↵ ↵ ⌘ ↵ Z Here ↵ is a continuous label that includes the spacetime coordinates. dµ↵ is a suitable measure on ↵-space, whereas ✏↵ is a function in ↵. When there is no room for confusion, we will omit the integration measure in our notation. The gauge variation δ↵ acts on composite fields by Leibniz’s rule as a di↵erentiation, and obey an (infinite dimensional) Lie algebra γ [δ↵,δβ]=f↵ δγ, (4.2) γ 7 where the structure constants f↵ by assumption are field independent. Formally, the partition function can be expressed as the Euclidean path integral S[φ] Z = [Dφ] e− , (4.6) Z up to some unspecified normalization of the integration measure [Dφ]. To define the latter properly, we can fix the gauge by replacing (4.6) with A A S[φ] Z = [Dφ] δ(F [φ]) det(δ↵F [φ])e− , (4.7) Z 7As an example, suppose the gauge symmetry of interest is di↵eomorphism. In this case, ↵ will be a ↵ ⇢ d spacetime coordinate y together with a vector index ⇢, ✏ will be a vector field " (y), with dµ↵ = d y.A scalar field ' transforms as δ '(x)= δd(x y)@ '(x). (4.3) y,⇢ − − ⇢ (4.1) then reproduces the usual infinitesimal di↵eomorphism '(x)= "⇢(x)@ '(x). From the action of the − ⇢ commutator of a pair of di↵eomorphisms on ', d d [δ ,δ ]'(x)=δ (x z)@ σ δ (x y)@ '(x) (y, ⇢ z,σ) y,⇢ z,σ − x − ⇢ − $ (4.4) = ddw δ⌫ δ⇥d(w z)@ δd(x ⇤y)+δ⌫ δd(w y)@ δd(x z) δ '(x), − ⇢ − σ − σ − ⇢ − w,⌫ Z ⇥ ⇤ we deduce the structure constant of di↵eomorphism, f (w,⌫) = δ⌫ δd(w z)@ δd(x y)+δ⌫ δd(w y)@ δd(x z). (4.5) (y,⇢)(z,σ) − ⇢ − σ − σ − ⇢ − 58 where A is an index that labels components of the gauge condition F A[φ]=0,suchthat A ↵ the Jacobian δ↵F [φ] is non-degenerate. We can then introduce the ghosts bA, c ,anda Lagrangian multiplier field BA,sothat(4.7)canbewrittenas Z = [DφDB Db Dc↵]exp S[φ]+iB F A[φ] b c↵δ F A[φ] . (4.8) A A − A − A ↵ Z In the full gauge fixed action with the ghosts included, the gauge symmetry is replaced by afermionicglobalsymmetryδB,knownasBRSTsymmetry,thatactsonthefieldsas δ φ = ic↵δ φ, B − ↵ δBBA =0, (4.9) δBbA = BA, i δ c↵ = f ↵cβcγ. B 2 βγ 2 The definition is such that δB =0,and iB F A[φ] b c↵δ F A[φ]=δ ib F A[φ] . (4.10) A − A ↵ B A It follows that δB is a symmetry of the action appearing in (4.8). One may also construct a Noether current jB associated with BRST symmetry, and define a corresponding BRST charge Q ,suchthatδ = i Q , ,where , stands for B B { B ·}cl {· ·}cl the Poisson bracket. If the path integral measure is also BRST invariant, then QB can be 2 promoted to a Hermitian quantum operator, such that QB =0. Aphysicalstate will be defined to be one that is BRST-invariant, or BRST-closed, | i Q =0.Intermsofthewavefunctional [φ], this condition is Q [φ] iδ [φ]= B| i B ⌘ B 0. It follows that the transition amplitude between two physical states is invariant under deformations of the gauge condition F A[φ]=0.Toseethis,considerachangeF A A A ! F + δ0F .Wehave A ↵ φf S[φ]+δB (bAF ) A δ0 f Ufi i = [DφDBADbADc ] ⇤ [φf ]e− δB(bAδ0F ) i[φi]=0, (4.11) h | | i φi f Z where Ufi is the time evolution operator defined by the path integral with boundary condi- tions φ = φi at the initial time and φ = φf at the final time. Furthermore, shifting a physical state by a BRST-exact state Q χ for some χ | i B| i | i does not a↵ect transition amplitudes between and any other physical states. Thus we | i may view as being equivalent to + Q χ , and identify the Hilbert space of physical | i | i B| i states with the cohomology of QB. 59 4.2 BRST on the worldsheet As already seen in section (2.4), the gauge fixing of the worldsheet theory of the critical bosonic string with the gauge condition gab =ˆgab leads to the action µ a µ a S[X ,bab,c ]=SP [ˆgab,X ]+Sgh[ˆgab,bab,c ], (4.12) In the language of BRST quantization, we have already integrated out the Lagrangian mul- tiplier field Bab as well as the dynamical metric gab.TheBRSTtransformationofbab, δBbab = Bab,isnowreplacedanoperatorthatisequivalenttoBab by the equation of motion, µ a which is the stress-energy tensor of the (X ,bab,c )system,namely X gh Tab = Tab + Tab . (4.13) The classical BRST transformation of the worldsheet fields, up to equations of motion, are µ a µ δBX = ic @aX , δBbab = iTab, (4.14) δ ca = icb ˆ ca. B rb B In the quantum theory, it is more precise to express this in terms of the BRST current ja as a local operator. In the conformal gaugeg ˆab = δab,workingincomplexcoordinates(z,z¯), the ghost fields can be written as b b , b b , and c cz, c cz¯,withthefollowing ⌘ zz ⌘ z¯z¯ ⌘ ⌘ singular OPE among holomorphic ghost fields e e 1 1 b(z)c(0) ,c(z)b(0) . (4.15) ⇠ z ⇠ z The bc system by itself is a CFT with holomorphic stress-energy tensor T gh = (@b)c 2b@c. (4.16) − − As usual, an implicit normal ordering is defined by subtracting o↵singular terms in the coincidence limit, and this form of the ghost stress-energy tensor is unambiguously fixed by the requirement that b is a primary of conformal weight (2, 0) and whereas c is one of weight ( 1, 0). One can indeed verify that the OPE of T gh with itself takes the form of the Virasoro − algebra with central charge cgh = 26. − Now we will denote the holomorphic and anti-holomorphic components of the BRST current by j j , j j ,andwritetheBRSTchargeas Bz ⌘ B Bz¯ ⌘ B dz dz¯ eQ = j (z) j (¯z). (4.17) B 2⇡i B − 2⇡i B I I e 60 Note that the holomorphic and anti-holomorphic BRST currents are independently con- served. The holomorphic BRST current is 3 j = cT X + bc@c + @2c, (4.18) B 2 X 1 µ µ where T = @X @Xµ is the stress-energy tensor of the X CFT, of central charge ↵0 − 2 B D.Thelasttermproportionalto@ c does not a↵ect QB,andisincludedsothatj can transform as a conformal primary. In fact, jB(z)isaconformalprimaryonlywhenD =26. QB obeys Q ,b = T = T X + T gh. (4.19) { B } The nilpotency of QB implies that T (z)T (0) = T (z) Q ,b(0) = Q ,T(z)b(0) (4.20) { B } { B } has only 1 singularity, and thus the total central charge c = D 26 must be zero. We z2 − will see later that this is also the condition for the cancelation of Weyl anomaly on a curved worldsheet. For computing BRST cohomology, it is useful to work with the oscillator representation of the BRST charge, X m n QB = cnL n + − : cmcnb m n : c0 +(anti holomorphic), (4.21) − 2 − − − − n Z m,n Z X2 X2 where the normal ordering prescription is defined by moving all the positively graded os- cillators to the right of the negatively graded ones, including the appropriate signs when a pair of Grassmannian fields are exchanged.8 One can also directly verify from (4.21) that Q ,b = LX + Lgh. { B n} n n Let us comment on the space of states in the bc ghost system. The space of local vertex operators are generated by the OPE of b, c, and their derivatives. However, the lowest weight operators are not the identity operator, but c and c@c, both of which has conformal weight 1. In the Fock space description, they correspond to two degenerate ground states and − |#i ,thatobey |"i b0 = c0 =0, |#i |"i (4.22) c = ,b = . 0|#i |"i 0|"i |#i Indeed, under the state/operator mapping, corresponds to the operator c(0), whereas |#i b 1 is mapped to the identity operator. Combing left and right movers, we will write e.g. − |#i , for the state that corresponds to the operator cc(0). |# #i 8Note that this normal ordering on the Fourier modes is a priori unrelated to the normal ordering prescription on the products of local field operators. e 61 The bc system has a U(1) ghost number symmetry that assigns (left-moving) ghost num- ber 1 to c and 1tob. The corresponding Noether current is − j = bc. (4.23) gh − While jgh is a conserved current of weight (1, 0), it is not a Virasoro primary. Indeed, the gh OPE of T with jgh takes the form 3 1 1 T gh(z)j (0) + j (0) + @j (0). (4.24) gh ⇠z3 z2 gh z gh This leads to an anomaly in the ghost number symmetry on a curved worldsheet, namely the operator jgh has a nonzero divergence 3 ja = R(g), (4.25) ra gh 4 where R(g)istheRicciscalaroftheworldsheetmetricgab.Consequently,thecorrelation functions of the bc CFT on a genus g Riemann surface must violate ghost number by 3 − 3g.Thereisasimilarviolationoftheghostnumberintheanti-holomorphicbc CFT.
Load more

Recommended publications
  • String Theory. Volume 1, Introduction to the Bosonic String
    This page intentionally left blank String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal field theory. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactification and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the finiteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersym- metric string theories and goes on to a broad presentation of the important advances of recent years. The first three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal field theory. The final four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries.
    [Show full text]
  • Perturbative Renormalization and BRST
    Perturbative renormalization and BRST Michael D¨utsch Institut f¨ur Theoretische Physik Universit¨at Z¨urich CH-8057 Z¨urich, Switzerland [email protected] Klaus Fredenhagen II. Institut f¨ur Theoretische Physik Universit¨at Hamburg D-22761 Hamburg, Germany [email protected] 1 Main problems in the perturbative quantization of gauge theories Gauge theories are field theories in which the basic fields are not directly observable. Field configurations yielding the same observables are connected by a gauge transformation. In the classical theory the Cauchy problem is well posed for the observables, but in general not for the nonobservable gauge variant basic fields, due to the existence of time dependent gauge transformations. Attempts to quantize the gauge invariant objects directly have not yet been completely satisfactory. Instead one modifies the classical action by adding a gauge fixing term such that standard techniques of perturbative arXiv:hep-th/0411196v1 22 Nov 2004 quantization can be applied and such that the dynamics of the gauge in- variant classical fields is not changed. In perturbation theory this problem shows up already in the quantization of the free gauge fields (Sect. 3). In the final (interacting) theory the physical quantities should be independent on how the gauge fixing is done (’gauge independence’). Traditionally, the quantization of gauge theories is mostly analyzed in terms of path integrals (e.g. by Faddeev and Popov) where some parts of the arguments are only heuristic. In the original treatment of Becchi, Rouet and Stora (cf. also Tyutin) (which is called ’BRST-quantization’), a restriction to purely massive theories was necessary; the generalization to the massless case by Lowenstein’s method is cumbersome.
    [Show full text]
  • 1 Hamiltonian Quantization and BRST -Survival Guide; Notes by Horatiu Nastase
    1 Hamiltonian quantization and BRST -survival guide; notes by Horatiu Nastase 1.1 Dirac- first class and second class constraints, quan- tization Classical Hamiltonian Primary constraints: φm(p; q) = 0 (1) imposed from the start. The equations of motion on a quantity g(q; p) are g_ = [g; H]P:B: (2) Define φm ∼ 0 (weak equality), meaning use the constraint only at the end of the calculation, then for consistency φ_m ∼ 0, implying [φm; H]P:B: ∼ 0 (3) The l.h.s. will be however in general a linearly independent function (of φm). If it is, we can take its time derivative and repeat the process. In the end, we find a complete set of new constrains from the time evolution, called secondary constraints. Together, they form the set of constraints, fφjg; j = 1; J. A quantity R(q; p) is called first class if [R; φj]P:B: ∼ 0 (4) for all j=1, J. If not, it is called second class. Correspondingly, constraints are also first class and second class, independent of being primary or sec- ondary. To the Hamiltonian we can always add a term linear in the constraints, generating the total Hamiltonian HT = H + umφm (5) where um are functions of q and p. The secondary constraint equations are [φm; HT ] ∼ [φm; H] + un[φm; φn] ∼ 0 (6) where in the first line we used that [un; φm]φn ∼ 0. The general solution for um is um = Um + vaVam (7) with Um a particular solution and Vam a solution to Vm[φj; φm] = and va arbitrary functions of time only.
    [Show full text]
  • M-Theory Solutions and Intersecting D-Brane Systems
    M-Theory Solutions and Intersecting D-Brane Systems A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy in the Department of Physics and Engineering Physics University of Saskatchewan Saskatoon By Rahim Oraji ©Rahim Oraji, December/2011. All rights reserved. Permission to Use In presenting this thesis in partial fulfilment of the requirements for a Postgrad- uate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professor or professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of material in this thesis in whole or part should be addressed to: Head of the Department of Physics and Engineering Physics 116 Science Place University of Saskatchewan Saskatoon, Saskatchewan Canada S7N 5E2 i Abstract It is believed that fundamental M-theory in the low-energy limit can be described effectively by D=11 supergravity.
    [Show full text]
  • String Field Theory, Nucl
    1 String field theory W. TAYLOR MIT, Stanford University SU-ITP-06/14 MIT-CTP-3747 hep-th/0605202 Abstract This elementary introduction to string field theory highlights the fea- tures and the limitations of this approach to quantum gravity as it is currently understood. String field theory is a formulation of string the- ory as a field theory in space-time with an infinite number of massive fields. Although existing constructions of string field theory require ex- panding around a fixed choice of space-time background, the theory is in principle background-independent, in the sense that different back- grounds can be realized as different field configurations in the theory. String field theory is the only string formalism developed so far which, in principle, has the potential to systematically address questions involv- ing multiple asymptotically distinct string backgrounds. Thus, although it is not yet well defined as a quantum theory, string field theory may eventually be helpful for understanding questions related to cosmology arXiv:hep-th/0605202v2 28 Jun 2006 in string theory. 1.1 Introduction In the early days of the subject, string theory was understood only as a perturbative theory. The theory arose from the study of S-matrices and was conceived of as a new class of theory describing perturbative interactions of massless particles including the gravitational quanta, as well as an infinite family of massive particles associated with excited string states. In string theory, instead of the one-dimensional world line 1 2 W. Taylor of a pointlike particle tracing out a path through space-time, a two- dimensional surface describes the trajectory of an oscillating loop of string, which appears pointlike only to an observer much larger than the string.
    [Show full text]
  • Free Lunch from T-Duality
    Free Lunch from T-Duality Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D{07743 Jena, Germany [email protected] We consider a simple method of generating solutions to Einstein gravity coupled to a dilaton and a 2-form gauge potential in n dimensions, starting from an arbitrary (n m)- − dimensional Ricci-flat metric with m commuting Killing vectors. It essentially consists of a particular combination of coordinate transformations and T-duality and is related to the so-called null Melvin twists and TsT transformations. Examples obtained in this way include two charged black strings in five dimensions and a finite action configuration in three dimensions derived from empty flat space. The latter leads us to amend the effective action by a specific boundary term required for it to admit solutions with positive action. An extension of our method involving an S-duality transformation that is applicable to four-dimensional seed metrics produces further nontrivial solutions in five dimensions. 1 Introduction One of the most attractive features of string theory is that it gives rise to gravity: the low-energy effective action for the background fields on a string world-sheet is diffeomor- phism invariant and includes the Einstein-Hilbert term of general relativity. This effective action is obtained by integrating renormalization group beta functions whose vanishing is required for Weyl invariance of the world-sheet sigma model and can be interpreted as a set of equations of motion for the background fields. Solutions to these equations determine spacetimes in which the string propagates.
    [Show full text]
  • String Theory on Ads3 and the Symmetric Orbifold of Liouville Theory
    Prepared for submission to JHEP String theory on AdS3 and the symmetric orbifold of Liouville theory Lorenz Eberhardt and Matthias R. Gaberdiel Institut f¨urTheoretische Physik, ETH Zurich, CH-8093 Z¨urich,Switzerland E-mail: [email protected], [email protected] Abstract: For string theory on AdS3 with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows 3 4 us to show that the CFT dual of superstring theory on AdS3 × S × T for generic NS-NS flux is the symmetric orbifold of (N = 4 Liouville theory) × T4. For the case of minimal flux (k = 1), the Liouville factor disappears, and we just obtain the symmetric orbifold of T4, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on AdS3 × X. arXiv:1903.00421v1 [hep-th] 1 Mar 2019 Contents 1 Introduction1 2 Bosonic strings on AdS3 3 2.1 The sl(2; R)k WZW model and its free field realisation3 2.2 Vertex operators4 2.3 The DDF operators5 2.4 The identity operator8 2.5 The moding of the spacetime algebra8 2.6 Identifying Liouville theory on the world-sheet 11 2.7 Discrete representations 13 3 4 3 A review of superstrings on AdS3 × S × T 15 3.1 The RNS formalism 15 3.2 The hybrid formalism 17 3.3 Supergroup generators 18 4 The psu(1; 1j2)k WZW model 19 4.1 Wakimoto representation of sl(2; R)k+2 and vertex operators 19 4.2 The short representation 20 4.3 Spectral
    [Show full text]
  • Introduction to Superstring Theory
    Introduction to Superstring theory Carmen Nunez Instituto de Astronomia y Física del Espacio C.C. 67 - Sue. 28, 1428 Buenos Aires, Argentina and Physics Department, University of Buenos Aires [email protected] Abstract 1 Overview The bosonic string theory, despite all its beautiful features, has a number of short­ comings. The most obvious of these are the absence of fermions and the presence of tachyons in spacetime. The tachyon is not an actual physical inconsistency; it indicates at least that the calculations are being performed in an unstable vacuum state. More­ over, tachyon exchange contributes infrarred divergences in loop diagrams and these divergences make it hard to isolate the ultraviolet behaviour of the "unified quan­ tum theory" the bosonic string theory gives rise to and determine whether it is really satisfactory. Historically, the solution to the tachyon problem appeared with the solution to the other problem, the absence of fermions. The addition of a new ingredient, supersym- metry on the world-sheet, improves substantially the general picture. In 1977 Gliozzi, Scherk and Olive showed that it was possible to get a model with no tachyons and with equal masses and multiplicities for bosons and fermions. In 1980, Green and Schwarz proved that this model had spacetime supersymmetry. In the completely consisten- t tachyon free form of the superstring theory it was then possible to show that the one-loop diagrams were completely finite and free of ultraviolet divergences. While most workers on the subject believe that the finiteness will also hold to all orders of perturbation theory, complete and universally accepted proofs have not appeared so far.
    [Show full text]
  • Propagators and Path Integrals
    NAT10NAAI. WSTITUUT VOOR KERNFYSICA EN NIKHEF-H/95-050 Propagators and Path Integrals J.W. van Holten NIKHEF-H, P.O. Box 41882 1009 DB Amsterdam NI, August 22, 1995 Abstract Path-integral expressions for one-particle propagators in scalar and ferinioaic field theories are derived, for arbitrary mass. This establishes a direct connection between field theory and specific classical point-particle models. The role of world-line rep&rametroation invaiiance of the classical action avnd the implementation of the corresponding BB5T-symmetry in the qutntuin theory are discussed. The presence of classical world-line supersymrnetry is shown to lead t/> an unwanted doubling of states for massive spin-1/2 particles. The origin of this phenomenon is traced to a 'hidden' topologies! fermionic excitation. A different formulation of the pseudo-classical mechan- ics using a bosonic representation of 71 is shown to remove those extra states at the expense of losing manifest supersymmetry. PilKHEf SECTIE - H POSTBUS 4188',!, 1009 DB A^STERDfW •UL ? 0 NJ 0 1 KS002045549 R: FI NL96FG754 DEOOSS87924 1 Introduction Because of their conceptual simplicity, path-integral methods [1, 2] often provide convenient procedures to obtain insight in field theoretical problems. In recent work by Strassler, McKeon, Schmidt, Schubert and others [3]-[7] world-line path integrals have been applied to a reformulation of standard Feynman perturbation theory from which useful information on the structure of perturbative Green's functions is extracted. Some of these results were actually first derived in the particle-limit of string theory [8]. A basic question in this context is the representation of propagators in quantum field theory by path integrals for relativistic particles of various kind.
    [Show full text]
  • BRST Quantization of Yang-Mills Theory: a Purely Hamiltonian Approach on Fock Space
    PHYSICAL REVIEW D 97, 074006 (2018) BRST quantization of Yang-Mills theory: A purely Hamiltonian approach on Fock space Hans Christian Öttinger* ETH Zürich, Department of Materials, Polymer Physics, HCP F 47.2, CH-8093 Zürich, Switzerland (Received 1 March 2018; published 3 April 2018) We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories in an explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stage of the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilation operators, thus enabling us to characterize the physical states on a well-defined Fock space. The Hamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimal invariant extension of the free theory. It is verified that the evolution equations implied by the resulting minimal Hamiltonian provide a quantum version of the classical Yang-Mills equations. The modifications and requirements for the inclusion of matter are discussed in detail. DOI: 10.1103/PhysRevD.97.074006 I. INTRODUCTION The purpose of this paper is to show in the context of BRST quantization is a pivotal tool in developing Yang-Mills theories how all the above facets can be theories of the fundamental interactions, where the acro- handled entirely within the Hamiltonian approach, where nym BRST refers to Becchi, Rouet, Stora [1] and Tyutin explicit constructions on a suitable Fock space allow for a [2]. This method for handling constraints in the quantiza- maximum of intuition. The focus on Fock space implies a (quantum) particle interpretation rather than a field tion of field theories usually requires a broad viewpoint idealization.
    [Show full text]
  • Bosonic Strings: a Mathematical Treatment
    AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor Bosonic Strings: A Mathematical Treatment Jürgen Jost American Mathematical Society • International Press Selected Titles in This Series 21 J¨urgen Jost, Bosonic Strings: A Mathematical Treatment, 2001 20 Lo Yang and S.-T. Yau, Editors, First International Congress of Chinese Mathematicians, 2001 19 So-Chin Chen and Mei-Chi Shaw, Partial Differential Equations in Several Complex Variables, 2001 18 Fangyang Zheng, Complex Differential Geometry, 2000 17 Lei Guo and Stephen S.-T. Yau, Editors, Lectures on Systems, Control, and Information, 2000 16 Rudi Weikard and Gilbert Weinstein, Editors, Differential Equations and Mathematical Physics, 2000 15 Ling Hsiao and Zhouping Xin, Editors, Some Current Topics on Nonlinear Conservation Laws, 2000 14 Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, 2000 13 Vasilios Alexiades and George Siopsis, Editors, Trends in Mathematical Physics, 1999 12 Sheng Gong, The Bieberbach Conjecture, 1999 11 Shinichi Mochizuki, Foundations of p-adic Teichm¨uller Theory, 1999 10 Duong H. Phong, Luc Vinet, and Shing-Tung Yau, Editors, Mirror Symmetry III, 1999 9 Shing-Tung Yau, Editor, Mirror Symmetry I, 1998 8 J¨urgen Jost, Wilfrid Kendall, Umberto Mosco, Michael R¨ockner, and Karl-Theodor Sturm, New Directions in Dirichlet Forms, 1998 7 D. A. Buell and J. T. Teitelbaum, Editors, Computational Perspectives on Number Theory, 1998 6 Harold Levine, Partial Differential Equations, 1997 5 Qi-keng Lu, Stephen S.-T. Yau, and Anatoly Libgober, Editors, Singularities and Complex Geometry, 1997 4 Vyjayanthi Chari and Ivan B. Penkov, Editors, Modular Interfaces: Modular Lie Algebras, Quantum Groups, and Lie Superalgebras, 1997 3 Xia-Xi Ding and Tai-Ping Liu, Editors, Nonlinear Evolutionary Partial Differential Equations, 1997 2.2 William H.
    [Show full text]
  • BRST Formulation of the Gupta-Bleuler Quantization Method
    NL90C0266 MATIONAAL INSTITUUT VOOR KERNFYSICA EN HOCE-EMERGIEFYSICA March 1990 NIKHEF-H/90-7 BRST Formulation of the Gupta-Bleuler Quantization Method Z. Hastowicz *, J. Kowalski-Glikman \ J. Lukierski *. J.W, van Hotten t Abstract In this paper we show, how an algebra of mixed flits and second class constrain»» can ba transformed Into an algebra of the Gupta-Bleuler type, consisting of holomorphlc ar.d anti* hotomorphic constraints. We perform its quantization by BRST methods. We construct a second* level BRST operator n by Introducing a new ghost sector (the second-level ghosts), in addition to the ghosts of the standard BRST operator. We find an Inner product In this ghost sector such that the operator li ts hermltean. The physical states, as defined by the non-hermitsan holomorphlc constraints, are shown to be given in terms of the cohomology of this hermitean BRST charge. * Mstttuf tor TtwofHemt Pttyttca. UMvmrtlty of Wrodmw. ut Cybutokltgo «. Wnetmw. Poimnd • CHEAP, PO BOM 4IM?, 1009 DB Amstfdmm. Th» Nethftmnda. bit/ft rt00J)/fcf>«M.flSM»*itf f NSOHEF^ PO BOM 41*62. 1009 OB Amstontom. Thm Netherlands, bitnét 11ë&nikh*tli.nikh*t.nl NIKHEF SECTIE-H POSTBUS 41H3, 100» L«B AMSTERDAM 1 Introduction BRST-methods [1, 2} have become a powerful tool in the quantization of constrained systems. Not only do they provide a method for establishing the consistency of the covariant quantization procedure of gauge theories , such as used in the path-integral formulation of Yang-Mills theory [3], they also have become important in the elucidation of the canonical structure of constrained systems [4, 5, 6, 7] like relativistic point-particles and strings |8], In spite of all the past successes of the method, there still doesn't exist a completely unified and unique prescription for the application of BUST' methods to arbitrary constrained systems and surprises keep turning up when studying new systems with new types of gauge-invariances.
    [Show full text]