Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 49 (2016)

Dipolar quantization and the infinite circumference limit of 2d CFT†
T. Tada∗1We elaborate on our previous work1) , where a newterm dipolar quantization was introduced, and further
argue that adopting L0 − (L1 + L−1 )/2 as the Hamiltonian yields an infinite circumference limit in twodimensional conformal field theory. One of the physical significances of this new Hamiltonian is that thetime translational vector field coincides with the electric field of an electric dipole located at z = 1, as opposed to in the ordinary radial quantization case. The
new theory then exhibits a continuous and strongly
degenerated spectrum as well as the Virasoro algebrawith a continuous index.
First, let us define the following “charges”:1t
dzg(z)fκ (z)T (z),(1)
Lκ ≡
2πi

where T (z) = Tzz (z) is the holomorphic part of theenergy-momentum tensor of the original CFT. Further, the integration is performed along the path wherethe Euclidean time t assumes a constant value. The
relationship between the Euclidean time t and the z
coordinate is best summarized by introducing a newcoordinate w as
w = t + is,(2)

where s represents the Euclidean space part. Then,the following relations are imposed:∂w
1=,∂zg(z)fκ (z) = eκw .

(3)
(4)

Now we can define Lκ by choosing an appropriate valueof g(z). In particular, for κ = 0, we have the expression
for the Hamiltonian for g(z) below corresponding to
the case for the ordinary radial quantization and forthe dipolar quantization:
1tdzg(z)T (z)L0 =2πifor g(z) = z L0.
(5)=

L1 +L−1(z−1)2for g(z) = − 2
L0 −2
One can then calculate the commutation relations
between Lκ ’s as follows:+†∗1[Lκ , Lκ ] = (κ − κ )Lκ+κ
 c 3
 12 (κ − κ)δκ+κ ,0 for g(z) = zc 312 κ δ(κ + κ )2for g(z) = − (z−1)
2. (6)

2, κ assumes all the real valHere, for g(z) = − (z−1)
2ues, while κ is an integer for g(z) = z. By using thiscommutation relation with the Hamiltonian
H = L0 + L̄0 ,(7)|α, σ,(8)

L0 |α, σ = α|α, σ.

(9)Lκ |α, σ = |α − κ, σ.

(10)L†κ = L−κ ,(11)(L−1 )† = L−1 , (L0 )† = 2L−1 − L0 ,(L1 )† = L1 − 4L0 + 4L−1 .

(12)where L̄0 stands for a member of another set of similar“charges”, one can construct a state with an arbitraryvalue of energy in the following manner. First, consider
an eigenstate of L0 with an eigenvalue α and with anadditional index σ labeling a possible degeneracy:
so thatIn this case, operating on |α, σ with Lκ yields

Thus, starting from the vacuum or any other energy
eigenstate, we can construct an eigenstate for L0 withan arbitrary eigenvalue because κ can assume any realvalue.
Further, one can show that
where † stands for the Hermitian conjugation. Oneof the idiosyncrasies we find in this formulation is adifferent inner product of the Hilbert space. For the2choice of g(z) = − (z−1), the following holds:
2Equation (13) shows that the operations of the Hermitian conjugation operations in dipolar quantizationon L−1 , L0 and L1 are closed among themselves. Inaddition, they definitely assume a different form fromthose in radial quantization. Nonetheless, if we compute the Hermitian conjugate for the combination
L0 −(L1 +L−1 )/2, which is the Hamiltonian for dipolarquantization, it proves to be Hermitian (in the senseof dipolar quantization):†
L1 + L−1L0 −
2
L1 − 4L0 + 4L−1 + L−1= 2L−1 − L0 −(13)
2
L1 + L−1.
= L0 −2Thus, we presented a new conformal symmetric quantum system with novel Hamiltonian and Hilbert space.
References
1) N. Ishibashi and T. Tada, J. Phys. A 48, no. 31, 315402(2015) [arXiv:1504.00138 [hep-th]].

Condensed from arXiv:1602.01190 [hep-th].
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