Introduction to string theory 2 - Quantization

Remigiusz Durka Institute of Theoretical Physics Wroclaw / 34

Table of content Introduction to Quantization Classical String Quantum String 2 / 34

Classical Theory In the classical mechanics one has dynamical variables called: coordinates (q ; q 2; q 3) = q i and momenta (p ; p 2; p 3) = p i These specify the state of a classical system. The canonical structure (also called symplectic structure) of classical mechanics consists of Poisson Brackets between these variables. In canonical coordinates (q i ; p j ) on the phase space, given two functions f (p i ; q i ; t ) and g(p i ; q i ; t ), Poisson bracket takes the form: ff ; gg = NX i=k @f @q k @g @p k @f @g @p k @q k fx i ; p j g = NX i=k ik jk = ij Time-evolution of some quantity in the Hamiltonian formulation: d A = fa; Hg dt 3 / 34

Quantization Canonical quantization earliest method used to build quantum mechanics (Paul Dirac), applied to a classical field theory it is also called second quantization. Road to quantum mechanics leads through: Uncertainty Principle x p 2 This algebraic structure corresponds to a generalization of the canonical structure of classical mechanics. In quantum mechanics dynamical variables (q i ; p j ) become operators (^x i ; ^p j ) acting on a Hilbert space of quantum states. Poisson bracket (more generally the Dirac bracket) is replaced by: commutators [^x i ; ^p j ] = ^x i ^p j ^p j ^x i = i ij [ : ; : ]! i f : ; : g 4 / 34

The states of a quantum system can be labelled by the eigenvalues of any operator: jx i is a state which is an eigenvector of ^A with eigenvalue x Notationally: ^A jx i = x jx i A particularly important example is the position basis, which is the basis consisting of eigenstates of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any state j'i is associated with a complex-valued function of three-dimensional space. The wavefunction of a state j'i is '(x ) = hx j'i 5 / 34

Different way of quantization A later development was the Feynman path integral, a formulation of quantum theory which emphasizes the role of superposition of quantum amplitudes. The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude. 6 / 34

Relativistic point particle The motion of a relativistic particle with mass m can be formulated as a variational problem. Action for point particle: S 0 = mc ds where space-time interval: Z ds 2 = g dx dx We minimize lenght of one-dimensional worldline drawn by a point particle moving in D-dimensional space-time. 7 / 34

Relativistic string The motion of a relativistic string with tension T in a curved D-dimensional space-time can be formulated as a variational problem. Z Z S P = T da = T 2 d 2 p hh ab g (X )@ a X ()@ b X () We minimize area of two-dimensional world sheet sweept out by the string in D-dimensional space-time. The space-time embedding of the string world sheet is described by functions X (; ) coming from solving Euler-Lagrange eq. for h ab : 2 T ab p S P T h h = ab 0 @ a X @ b X 2 h abh cd @ c X @ d X = 0 Equation of motion X = p h @ a ( p hh ab @ b X ) = 0 Action S P, traditionally called the Polyakov action, was discovered by Brink, Di Vecchia and Howe and by Deser and Zumino several years before Polyakov. 8 / 34

Equations of motion Using diffeomorphisms and Weyl transformation one can transform the action into the following form: R d 2 _X 02 2 X S P = T 2 R dd p ab g (X )@ ax ()@ bx () = T 2 where ab = diag( ; ) S P = T 2 Z d 2 _X 2 X 02 Condition T ab = 0 (refered as Virasoro constraints): T 0 = T 0 = 0! T 00 = T = 0! 0 = _ X X 0 0 = ( _ X 2 + X 02 ) Equation of motion for X : ( d 2 d 2 d 2 d 2 )X = 0 or ( X X ) = 0 9 / 34

From Lagrange to Hamiltonian formalism Lagrangian density: L = 4 0 (@X @ X @ X @ X ) Conjugate momenta: = Solution of X for closed string: L (@ X ) = 2 0 @ X X (; ) = X L ( + ) + X ( R ) r 0 X X L ( + ) = 2 x + 0 p ( + ) + i 2 n6=0 r X X R ( ) = 2 x + 0 p ( ) + i 0 2 n6=0 n ~ n e n n e 2in(+) 2in( ) 0 / 34

Poisson brackets fx (); ( 0 )g = ( 0 ) fx (); ( 0 )g = 0 f (); ( 0 )g = 0 Hamiltonian density fp ; x g = f m ; ng = im m+n ;0 f ~ m ; ~ ng = im m+n ;0 f m ; ~ ng = 0 H = ( _ X L) = Hamiltonian (note that ( n ) = n): 4 0 (@X @ X + @ X @ X ) Z 2 H = H()d = 0 X ( n n + ~ n ~n ) / 34

Mode expansions of the energy momentum tensor (T = @ X @ X ): +X T = 2ls 2 2im( L ) me T++ m= +X = 2ls 2 ~L me 2im(+) m= where the Fourier coefficients are the Virasoro generators L m = 2 X n= m n n ; ~L m = 2 Comparing with the Hamiltonian for a closed string: H = X X n= ( n n + ~ n ~n ) = 2(L0 + ~ L 0) ~ m n ~ n while for an open string H = 2 X n n = L0 2 / 34

Witt algebra of L m s: fl m ; L n g = i(m n)l m+n f L ~ m ; L ~ n g = i(m n) L ~ m+n fl m ; L ~ n g = 0 In the quantum theory one needs to resolve ordering ambiguities! Mass formula for the string Classically the vanishing of the energy-momentum tensor translates into the vanishing of all the Fourier modes: L m = 0 for m = 0; ; 2; ::: It can be used to derive an expression for the mass of a string: L 0 + ~L 0 = 0 2 M 2 + X n= ( n n + ~ n ~n ) M 2 c = 2 0 X m= X ( n n + ~ n ~n ) M o 2 = 0 m=( n n ) 3 / 34

Quantum String 4 / 34

Due to the constraints quantization of the bosonic string isn t simple. Ways to implement the constraints basically fall into two categories: (I) Solve the constraints by singling out some spacetime directions from others. As a result some oscillators become dependent on others and are not to be independently quantized. The Hilbert space then manifestly has positive norm. Lorentz invariance is not manifest. (II) Quantise all the oscillators, and then impose the constraints on the Hilbert space. Only those states satisfying the constraints will be treated as physical. The full Hilbert space will have negative-norm states. However the constrained subspace of Hilbert space will be positive. In this procedure we maintain manifestation of the Lorentz invariance. 5 / 34

The world-sheet theory can now be quantized by replacing Poisson brackets by commutators: [ : ; : ]! i f : ; : g. [x ; p ] = i [ m ; ~ n ] = 0 [ m ; n ] = [~ m ; ~ n ] = m m+n ;0 Lets define new operators: lowering a m = p m m ( n )y = n rising a y m = p m m Algebra satisfied by the modes is essentially the algebra of raising and lowering operators for quantum-mechanical harmonic oscillators: [a m ; a y n ] = mn 6 / 34

This structure is expected because free quantum fields are simply composed of infinitely many harmonic oscillators. The corresponding Hilbert space is therefore a Fock space spanned by products of states jni, for n = 0; ; 2; :::,which are built on a normalized ground state j0i annihilated by the lowering operators a n. h0j0i = hmjni = mn a n j0i = 0 jni = (ay ) n p n! j0i Level operator (number operator) a y ajni = njni In each sector of the theory, i.e. open, and closed left-moving and rightmoving, we get D independent families of such infinite sets of oscillators, one for each spacetime dimension = 0; ; :::; D. 7 / 34

The only subtlety in this case is that, because 00 =, the time components are proportional to oscillators with the wrong sign, i.e. jj mj0ijj 2 = m [a 0 m ; a 0y m ] = Such oscillators are potentially dangerous, because they create states of negative norm which can lead to an inconsistent, non-unitary quantum theory with negative probabilities. However, as we will see, T ab = 0 constraint eliminates the negative norm states from the physical spectrum. Imposing the mass-shell constraints will thereby lead to the construction of physical states and appearance of elementary particles in the quantum string spectrum 8 / 34

It also follows that the zero mode operators x and p obey the standard Heisenberg commutation relations. They may thereby be represented on the Hilbert space spanned by the usual plane wave basis so string state also carries momentum jki = e ikx of eigenstates of p Thus the Hilbert space of the string is built on the states jk ; 0i label by center of mass momentum k with relations: p jk ; 0i = k jk ; 0i a mjk ; 0i = 0 For closed strings, there is also an independent left-moving copy of the Fock space. 9 / 34

Normal Ordering In the quantum theory operators are defined to be normal-ordered, that is all rising operators goes to the left: L m = 2 X : m n n : ~L m = n= 2 X n= : ~ m n ~ n : Then all L m for m 6= 0 are all fine promoted to quantum operators. However, L 0 needs more careful definition, because n and n do not commute (actually, this is the only Virasoro operator for which normal-ordering matters): L 0 = 2 2 0 + +X n= n n + c 20 / 34

Total zero-point energy of the families of oscillators: c = D 2 2 Of course a = formally, but, as usual in QFT, it can be regulated to give a finite answer coresponding to the total zero-point energy of all harmonic oscillators. Consider zeta function defined as the infinite sum: (z ) = X n= X n= ; z 2 C; Re(z ) > n z n ( ) = " + 2 + 3 + 4 + :::" = 2 c = D 2 24 Factor (D 2) appears because after imposition of the physical constraints remains only (D 2) independent polarizations of the string embeding fields X 2 / 34

Since a constant appeared in expression for L 0, one must expect a constant to be added to L 0 in all formulas, in particular the Virasoro algebra. In the quantum theory the Virasoro generators satisfy the relation: [L m ; L n ] = (m n)l m+n D 2 m(m 2 ) m+n ;0 where D is dimension of the spacetime. The term proportional to D is a quantum effect. This means that it appears after quantization and is absent in the classical theory. This term is called a central extension, and D is called a central charge. The constant term on the right-hand side of these commutation relations is often called the "conformal anomaly", as it represents a quantum breaking of the classical conformal symmetry algebra. 22 / 34

Quantum spectrum 23 / 34

Physical String Spectrum We define the "physical states" jphysi of the full Hilbert space to be those which obey the Virasoro constraints T ab = 0: (L 0 a)jphysi = 0 for a = c = D 2 24 L n jphysi = 0 for n > 0 These constraints are just the analogs of the "Gupta-Bleuler prescription" for imposing mass-shell constraints in quantum electrodynamics. Note that because of the central term in the Virasoro algebra, it is inconsistent to impose these constraints on both L n and L n. 24 / 34

Open String Spectrum The constraint L 0 = a is then equivalent to the "mass-shell condition" X L 0 = 0 M 2 + n= n n! M 2 = (N a) 0 Ground State N = 0 The ground state has a unique realization whereby all oscillators are in the Fock vacuum, and is therefore given by jk ; 0i. The momentum k of this state is constrained by the Virasoro constraints to have mass-squared given by k 2 = M 2 = a 0 < 0 This state therefore describes a "tachyon", i.e. a particle which travels faster than the speed of light. So the bosonic string theory is a not a consistent quantum theory, because its vacuum has imaginary energy and hence is unstable. 25 / 34

First Excited Level N = Way to get N = is to excite the first oscillator modes once, a jk ; 0i. We are also free to specify a "polarization vector" for the state. jk ; i = jk ; 0i The Virasoro constraints give the energy: M 2 = ( a) 0 Furthermore, using the commutation relations we may compute L jk ; i = p 2 0 (k )( )jk ; 0i = p 2 0 (k )jk ; 0i and thus the physical state condition L jk ; i = 0 implies that the polarization and momentum of the state must obey (k ) = 0. 26 / 34

Consistency The cases a 6= turn out to be unphysical. They contain: tachyons ghost states of negative norm So we shall take: a = where a = D 2 24 which fixes the bosonic critical dimension of spacetime D = 26 This condition can be regarded as the requirement of cancellation of the conformal anomaly in the quantum theory, obtained by demanding the equivalence of the quantizations in both the light-cone and conformal gauges. 27 / 34

Open String States Ground State N = 0 leads to: tachyon k 2 = M 2 = 0 < 0 The N = state jk ; i constructed with: k 2 = M 2 = 0 k = 0 has D 2 = 24 independent polarization states, as the physical constraints remove two of the initial D vector degrees of freedom. First excitied state N = describes: massless spin (vector) particle with polarization vector, which agrees with what one finds for a massless Maxwell foton. 28 / 34

Closed String Spectrum We now have to also incorporate the left-moving sector Fock states and a new condition now arises. Adding and subtracting the two physical state conditions (for a = ): (L 0 )jphysi = 0 ( ~ L 0 )jphysi = 0 (L 0 + ~ L 0 2)jphysi = 0 (L 0 ~L 0 )jphysi = 0 We arrive at the new mass-shell relation M 2 = 4 0 (N ) with N = ~ N 29 / 34

Ground State N = 0 The ground state is jk ; 0; 0i and it has mass-squared M 2 = 4 0 < 0 First Excited Level N = jk ; i = ( jk ; 0i ~ jk ; 0i) and it has a mass-squared M 2 = 0 The Virasoro constraints in addition give:l jk ; i) = L ~ jk ; i) = 0. which are equivalent to k = 0. 30 / 34

A polarization tensor obeying k = 0 encodes three distinct spin states according to the decomposition of into irreducible representations (irreducible symmetric, antisymmetric and trivial) of the spacetime "Little group" S 0(24), which classifies massless fields in this case. Tensor decompose into: = 2 ( + ) = 25 tr () g ) + + 2 ( ) + B + 25 tr () The symmetric, traceless tensor g corresponds to the spin 2 "graviton field" and it yields the spacetime metric. The antisymmetric spin 2 tensor B is called the "Neveu-Schwarz B-field" Scalar field is the spin 0 "dilaton". 3 / 34

Short Summary At Gound State (N = 0) for both strings we get: tachyon At the lowest level of massless states (N = ): open strings correspond to gauge theory closed strings correspond to gravity The higher levels N > 2 give an infinite tower of massive particle excitations. 32 / 34

So far we ve studied bosonic string theories, both open and closed: these string theories live in 26-dimensional spacetime all of their quantum states represent bosonic particle states among them we found important bosonic particles, such as the graviton and the photon non-abelian gauge bosons are needed to transmit the strong and weak forces and they too arise in bosonic string theory. Realistic string theories, however, must also contain the states of fermionic particles like quarks and leptons. to obtain them we need superstring theories critical spacetime dimension will be D = 0 we get rid off tachyonic states 33 / 34

Introduction to Quantisation Classical String Quantum String The end Introduction to string theory2 - Quantization 34 / 34