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The Quantized Fermionic String Class 2: The quantized fermionic string Canonical quantization Light-cone quantization Spectrum of the fermionic string, GSO projection . The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 . One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections . which result in the Type IIA and Type IIB superstring.. This is equivalent to the transition from classical mechanics to quantum mechanics via canonical commutation relations for the coordinates and their conjugate momenta. The Poisson brackets are promoted to (anti)commutators according to 1 1 [ ; ] ! [ ; ] ; f ; g ! f ; g PB i PB i Canonical quantization We will consider the functions X µ(τ; σ) and µ(τ; σ) as quantum mechanical operators . The Poisson brackets are promoted to (anti)commutators according to 1 1 [ ; ] ! [ ; ] ; f ; g ! f ; g PB i PB i Canonical quantization We will consider the functions X µ(τ; σ) and µ(τ; σ) as quantum mechanical operators This is equivalent to the transition from classical mechanics to quantum mechanics via canonical commutation relations for the coordinates and their conjugate momenta. Canonical quantization We will consider the functions X µ(τ; σ) and µ(τ; σ) as quantum mechanical operators This is equivalent to the transition from classical mechanics to quantum mechanics via canonical commutation relations for the coordinates and their conjugate momenta. The Poisson brackets are promoted to (anti)commutators according to 1 1 [ ; ] ! [ ; ] ; f ; g ! f ; g PB i PB i . Canonical quantization In this way we obtain for the equal time commutators h i X µ(σ); X_ µ(σ0) = 2πiα0ηµνδ(σ; σ0) [ ] h i X µ(σ); X µ(σ0) = X_ µ(σ); X_ µ(σ0) = 0 f µ ν 0 g f µ ν 0 g − 0 µν +(σ); +(σ ) = −(σ); −(σ ) = 2πδ(σ σ )η ; f µ ν 0 g +(σ); −(σ ) = 0 . The Fourier expansion coefficients are now operators with the following (anti)commutation relations µ ν µ ν µν [αm; αn] = [~αm; α~n] = mδm+nη ; µ ν [~αm; αn] = 0 ; [xµ; pν] = iηµν f µ νg f~µ ~νg µν br ; bs = br ; bs = δr+s η ; f µ ~νg br ; bs = 0 The reality conditions become hermiticity conditions: µ y µ µ y µ (αm) = α−m ; (~αm) =α ~−m µ p1 µ µ y p1 µ and if we rescale them as: am = m αm,(am) = m α−m for µ m > 0 , then the am satisfy the usual harmonic oscillator . commutation relations. Recall the bosonic string: The ground state is annihilated by all positive modes and we choose it to be an eigenstate of the center µ µ µ of mass momentum operator α0 andα ~0 with eigenvalue p : µ j µ µ j µ αm 0; p > = 0 ; α~m 0; p >= 0 ; for m > 0 p^µj0; pµ > = pµj0; pµ > To examine the states in the Hilbert space of the fermionic theory we have to distinguish between R and NS sectors. Hilbert space We define oscillators with positive (negative) mode numbers as annihilation (creation) operators. To examine the states in the Hilbert space of the fermionic theory we have to distinguish between R and NS sectors. Hilbert space We define oscillators with positive (negative) mode numbers as annihilation (creation) operators. Recall the bosonic string: The ground state is annihilated by all positive modes and we choose it to be an eigenstate of the center µ µ µ of mass momentum operator α0 andα ~0 with eigenvalue p : µ j µ µ j µ αm 0; p > = 0 ; α~m 0; p >= 0 ; for m > 0 p^µj0; pµ > = pµj0; pµ > . Hilbert space We define oscillators with positive (negative) mode numbers as annihilation (creation) operators. Recall the bosonic string: The ground state is annihilated by all positive modes and we choose it to be an eigenstate of the center µ µ µ of mass momentum operator α0 andα ~0 with eigenvalue p : µ j µ µ j µ αm 0; p > = 0 ; α~m 0; p >= 0 ; for m > 0 p^µj0; pµ > = pµj0; pµ > To examine the states in the Hilbert space of the fermionic theory we have to distinguish between R and NS sectors. Hilbert space The oscillator ground state in the NS sector is defined by 1 3 αµ j0 > = bµj0 > = 0 ; m = 1; 2;:::; r = ; ;::: m NS r NS 2 2 and the ground state in the R sector is defined as µ j µ j αm a >R = bm a >R = 0 ; m = 1; 2;:::; j µ a >R is because of the zero modes b0 . Recall f µ νg f~µ ~νg µν br ; bs = br ; bs = δr+s η ; f µ νg µν Then b0 ; b0 = η . µ 2 ) It is easy to check thatQ [b0 ; M ] = 0 = µi the states ja > and b ja > are degenerate. in. mass.. i 0 . Hilbert space The mass-shell condition is determined by the zero-frequency part of the Virasoro constraints. Recall the closed bosonic string X1 0 1 2 α 2 L = α− · α + α = N + p 0 n n 2 0 4 n=1 X1 0 1 2 α 2 L~ = α~− · α~ + α = N~ + p 0 n n 2 0 4 n=1 Then 2 − µ 2 2 M = p pµ = ML + MR ; 0 2 ~ 0 2 α ML = 2(N + a) ; α MR = 2(N + a) a is the normal ordering constant accounting for the ordering 2 2 ambiguity of L0 and M = M . L R . Normal ordering The super-Virasoro generators are undefined without giving an operator ordering prescription. We define them by normal ordering: annihilation operators to the right of creation operators L = L(α) + L(b) n nX n (α) 1 L = : α− · α : n 2 m m+n m2Z X ( ) (b) 1 n L = r + : b− · b : n 2 2 r n+r r2Z+ϕ>0 X Gr = α−m · br+m m2Z Ambiguity in L0 taken into account by normal ordering constant a ! a and a . NS R . Level numbers The mass operators for the fermionic string are given by the same expressions as in the bosonic case: α0M2 = N + constant but with level number operator: N = N(α) + N(b) X1 X (α) (b) N = α−m · αm N = rb−r · br m=1 r2Z+ϕ>0 and constant = aNS or aR . µ 0 2 An excitation by α−m with m > 0 increases α M by m units. µ 0 2 An excitation by b−r with r > 0 increases α M by r units. In the R sector the groundQ state is degenerate: j µi j The states 0 > and i b0 0 > are degenerate in mass.
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