Quantizing the relativistic string reveals one of the most stunning surprises in the history of theoretical physics: string theory is not a framework that can exist in any arbitrary spacetime dimensionality. Quantum consistency demands that the classical conformal anomaly must cancel, fixing the dimensionality of target spacetime to a rigid, unavoidable number. In this installment, we perform canonical Light-Cone Quantization, derive the Virasoro Algebra, uncover the critical dimension (D = 26), and extract the quantum spectrum of the graviton.
1. Light-Cone Gauge Fixing and Transverse Oscillators
In conformal gauge, residual conformal diffeomorphism freedom remains. To eliminate all unphysical unconstrained ghost states and isolate the true physical degrees of freedom, we define Light-Cone Coordinates for any spacetime vector (A^mu):
[
A^pm quiv rac{A^0 pm A^{D-1}}{sqrt{2}}, quad A^i quad (i = 1, dots, D-2)
]
The spacetime inner product becomes (A cdot B = -A^+ B^- – A^- B^+ + A^i B^i). We exploit the residual gauge invariance to set the Light-Cone Gauge:
[
X^+( au, sigma) = 2alpha’ p^+ au
]
Under this gauge choice, the non-linear Virasoro constraints (T_{ab} = 0) can be solved algebraically for (X^-), expressing (X^-) entirely in terms of the transverse coordinates (X^i( au, sigma)). The independent transverse physical coordinates expand into Fourier oscillator modes:
[
X^i( au, sigma) = x^i + 2alpha’ p^i au + isqrt{rac{alpha’}{2}} sum_{n
eq 0} rac{1}{n} left( alpha_n^i e^{-2in( au-sigma)} + ilde{alpha}_n^i e^{-2in( au+sigma)}
ight)
]
Canonical quantization imposes equal-time commutation relations on the oscillator modes:
[
[alpha_m^i, alpha_n^j] = m , delta^{ij} , delta_{m+n, 0}, quad [ ilde{alpha}_m^i, ilde{alpha}_n^j] = m , delta^{ij} , delta_{m+n, 0}
]
where (alpha_{-m}^i = (alpha_m^i)^dagger) acts as a creation operator and (alpha_m^i) ((m > 0)) as an annihilation operator.
2. The Virasoro Algebra and Central Charge Anomaly
The Fourier modes of the classical stress-energy tensor define the Virasoro Generators (L_m):
[
L_m = rac{1}{2} sum_{n=-infty}^infty alpha_{m-n} cdot alpha_n
]
Classically, the Poisson brackets satisfy the Witt algebra. At the quantum level, normal ordering ambiguities for (L_0) introduce an anomalous quantum mechanical central extension. Computing the commutator ([L_m, L_n]):
[
[L_m, L_n] = (m – n)L_{m+n} + rac{c}{12}(m^3 – m)delta_{m+n, 0}
]
where (c) is the Central Charge. For (D) free transverse scalar fields (X^mu), each field contributes (c_i = 1), giving (c_{ ext{matter}} = D). When gauge-fixing using the Faddeev-Popov path integral, the anti-commuting reparametrization ghost fields ((b, c)) contribute an intrinsic negative central charge (c_{ ext{ghost}} = -26). For the quantum string path integral to be anomaly-free and preserve Weyl gauge invariance:
[
c_{ ext{total}} = c_{ ext{matter}} + c_{ ext{ghost}} = D – 26 = 0 quad implies quad D = 26
]
Thus, bosonic string theory is quantum mechanically consistent if and only if spacetime has precisely 26 dimensions.
3. The Mass Spectrum and the Inevitability of the Graviton
The mass-shell condition for a closed bosonic string is governed by the Virasoro zero-modes (L_0) and ( ilde{L}_0):
[
M^2 = rac{4}{alpha’} (N – a) = rac{4}{alpha’} ( ilde{N} – a)
]
where (N = sum_{n=1}^infty alpha_{-n}^i alpha_n^i) is the transverse level operator, and (a) is the normal ordering Casimir zero-point energy summed over (D-2 = 24) transverse bosonic modes:
[
a = -rac{D-2}{2} sum_{n=1}^infty n = -rac{24}{2} left(-rac{1}{12}
ight) = 1
]
Evaluating the lowest mass eigenstates:
- Level (N = ilde{N} = 0) (Ground State):
[
M^2 = rac{4}{alpha’}(0 – 1) = -rac{4}{alpha’} < 0
]
This is a Tachyon ((M^2 < 0)). As Ashoke Sen proved, a tachyon does not imply faster-than-light particles; it signifies that bosonic string theory is expanded around the unstable apex of a scalar potential, rolling toward true tachyon condensation. - Level (N = ilde{N} = 1) (First Excited State):
[
M^2 = rac{4}{alpha’}(1 – 1) = 0
]
The state is created by acting with left and right creation operators on the vacuum: (|e^{ij}
angle = alpha_{-1}^i ilde{alpha}_{-1}^j |0
angle). Decomposing the (24 imes 24 = 576) states under the transverse rotation group ( ext{SO}(24)):
[
mathbf{24} otimes mathbf{24} = mathbf{299}_{ ext{symmetric traceless}} oplus mathbf{276}_{ ext{antisymmetric}} oplus mathbf{1}_{ ext{trace}}
]- Graviton ((h_{mu
u})): Massless, symmetric, traceless spin-2 tensor field. - Kalb-Ramond Field ((B_{mu
u})): Massless antisymmetric 2-form tensor gauge field. - Dilaton ((Phi)): Massless scalar field whose vacuum expectation value sets the string coupling constant: (g_s = e^{langle Phi
angle}).
- Graviton ((h_{mu
Here lies the crowning glory of string theory: physicists did not set out to invent quantum gravity. The massless spin-2 graviton emerges automatically and unavoidably from the quantum harmonics of a closed string.
