2.2 String theory
2.2.2 The quantum version
dσ
= 1 2
∞
X
n=−∞
αm−n· αn
= 0. (2.20)
2.2.2 The quantum version
The first quantization of open bosonic strings is presented in this subsection. It is known that one standard method of getting into the quantum physics from classical is to promote the physical quantities and Fourier modes to operators. This is equivalent by the substi-tution: replace the classical Poisson brackets with commutator, i.e. [· · · ]P.B. → −i[· · · ].
Thus, equation (2.18) and (2.19) need to be rewritten as follows
[Xµ(σ), Πν(σ0)] = iηµνδ(σ − σ0),
[αµm, ανn] = mδm+nηµν, (2.21) [xµ, pν] = iηµν.
2.2. STRING THEORY
Similar analogy are able to be made with simple harmonic oscillators in quantum mechanics. If we normalize αµm’s such that aµm ≡ αµm/√
m, then [aµm, aνn†] = ηµνδm−n. The physical interpretation of aµm’s is also very much similar to that in simple harmonic oscillators. For αµm with m > 0, it lowers a physical state and as a result aµm>0|0i = 0.
In contrast, an operator aµ−m with m > 0 rises the level of a physical state. Since the world sheet contains momentum which does not share the same Hilbert space with the oscillation operators. Therefore, a completely ground state for an bosonic open string can be denoted as |0; pi satisfies
αµm>0|0; pi = 0, (2.22)
pˆµ|0; pi = pµ|0; pi. (2.23)
The constraints of the classical theory correspond to the vanishing of the energy momen-tum tensors T++ and T−− as shown in equation (2.20). In quantum level, the vanishing of Lm in classical theory should be replaced by the positive frequency modes annihilate a physical state |ψi, that is
Lm>0|ψi = 0. (2.24)
This is much like the Gupta–Bleuler treatment in quantizing the E.M. theory. But L0 should be discussed independently since there exists an ordering ambiguity due to normal ordering. The normal–ordered expression of L0 is
L0 = 1 2α20+
∞
X
n=1
α−n· αn (2.25)
up to a to–be–determined constant say a. We include a and demand that a physical state
|ψi must satisfy
(L0− a)|ψi = 0. (2.26)
Choose a = 1 to avoid ghosts. In addition, equation (2.26) carries the information of mass M of open strings. From M2 = −p2 and the number operator N ≡P
kα−k· αk, we have M2 = − 2 + 2
∞
X
k=1
α−k · αk
= 2(N − 1). (2.27)
2.2. STRING THEORY
For example, the scalar ground state (N = 0) with M2 = −2 is tachyon. In contrast, the first excited state N = 1 is given by · α−1|0; pi , which has M2 = 0 and thus is a massless vector particle with polarization .
Equation (2.24) and (2.26) form the essential conditions for physical states. Lm and L0are the so–called Virasoro generators of bosonic open strings satisfying the the Virasoro algebra
[Lm, Ln] = (m − n)Lm+n+D(m3− m)
12 δm+n, (2.28)
where D means the dimension of the space–time, which is 26 if we choose a = 1.
Chapter 3
Spinor semiology and the application of BCFW rercursion relation
In this Chapter, a brief introduction to the spinor notations [9–11] is given by solving Dirac equation of a massless particle. After these kind of notations have been introduced, some Lorentz invariant quantities are created in terms of these notations. Having the above preparations, those quantities will be used to build up the 4–gluon scattering amplitudes by employing the BCFW technique. This calculation is provided in the last section.
3.1 Spinor notations
In this section, we would like to introduce the spinor notations. Throughout the whole section, the Lorentz signature ηµν = diag(−1, 1, 1, 1) is used. At first, consider a spin-1/2 particle with momentum p. Its behaviors can be understood by solving the Dirac equation
γ · p ψ(p) = 0 (3.1)
with p2 = 0. The gamma matrices in the Dirac representation (or standard representation) are
γ0
D =
I 0 0 −I
, (γi)D =
0 σi
−σi 0
(3.2)
3.1. SPINOR NOTATIONS
Those are the well–known Pauli matrices.
In order to distinguish the Dirac solutions from the Weyl solutions, we add “W ” to denote the case in Weyl representation and “D” for Dirac. The solution of Dirac equation (3.1) are often written as
ψ(p) =
Expanding equation (3.1) yields
−p0ψA+ ~σ · ~pψB = 0, −~σ · ~pψA+ p0ψB = 0. (3.5)
Above equations (3.5) give us two choices for the solutions of ψA and ψB. They are respectively ψA= ψB, ψA= −ψB. For positive energy p0 > 0, we have
For the case of ψA= −ψB, the corresponding wave function should be
ψA = −ψB⇒ ψ−(p) = 1
3.1. SPINOR NOTATIONS
Besides the Dirac representation, the so–called Weyl representation is also common to see.
In Weyl representation, the gamma matrices are
γµ=
0 − (¯σµ)αβ˙
− (σµ)α ˙β 0
. (3.9)
σµ and ¯σµ are defined in the following
σµ = (I, ~σ) , σ¯µ= (I, −~σ) . (3.10)
~σ are still the three Pauli matrices.
The off-diagonal gamma matrices in (3.9) imply ψ(p) could be divided into the com-bination of two 2-component spinors obey different kinds of transformation, i.e.
ψ(p) =
ξβ˙ ηα
. (3.11)
The lower undotted index α and the upper dotted one ˙β label the components of spinors η and ξ with both of the indices running from 1 to 2. The transformations of η and ξ are assigned in the following: If we denote ˜η = η†, ξ = ξ˜ †, and the SL(2, C) transformation matrix A with det(A) = 1, then we have
η0 = Aη → ηα, (3.12)
˜
η0 = (η0)† = ˜ηA†→ ˜ηα˙, (3.13)
ξ0 = ξA−1 → ξα, (3.14)
ξ˜0 = (ξ0)†= A−1†ξ → ˜˜ ξα˙. (3.15)
The consistency of the index structure implies following indices assignments for the transformation matrices:
A → Aαβ, A†→ A†β˙
˙ α, A−1 → A−1
β
α, A−1†
→
A−1†α˙ β˙
(3.16)
3.1. SPINOR NOTATIONS
For the reason that the Lagrangian of Dirac equation have to be Lorentz invari-ant, σµ, ¯σµ, A and the Lorentz transformation Lµν are required to satisfy the following relations:
AσµA†= Lµνσν (3.17)
A†−1
¯
σµA−1 = Lµνσ¯ν. (3.18)
Consistency requires the following indices assignments for σµ and ¯σµ, i.e.
σµ→ (σµ)α ˙β, σ¯µ→ ( ¯σµ)αβ˙ . (3.19)
If we define a 2 × 2 matrix as the following
≡ iσ2 =
0 1
−1 0
= −−1. (3.20)
where σ2 is the Pauli matrix. We can soon find out, by direct calculation, that
(σµ)T −1 = ¯σµ. (3.21)
If we add the indices into the above equation (3.21), it leads to
α ˙˙γ(σµ)γδ˙ −1δβ
= (¯σµ)αβ˙
⇒ (σµ)βα˙ = −1
β ˙˙γ(¯σµ)γδ˙ δα. (3.22)
From equation (3.22), we can immediately see that changes a upper undotted (a lower dotted) index into an lower undotted (an upper dotted). In contrast to , −1 changes an upper dotted (a lower undotted ) index into a lower dotted (an upper undotted) . Therefore, the index structures for and − are
αβ = α ˙˙β, −1αβ
= −1
˙
α ˙β (3.23)
such that
ηα = αβηβ, ηα = (−1)αβηβ. (3.24)
and −1 here are very much similar to the metric tensor gµν.
3.1. SPINOR NOTATIONS
Expanding the Dirac equation (3.1), η and ξ satisfy
~σ · ~p
|~p| ξ = −ξ, ~σ · ~p
|~p| η = η. (3.25)
For positive energy |~p| = p0 > 0, the equation for η means the solution has positive helicity or right–handed while negative helicity or left–handed for ξ. The Weyl represen-tation is related to Dirac represenrepresen-tation by a similarity transformation
(γµ)W = S (γµ)DS−1, where S is the transformation matrix given by
S = 1
This transformation also implies that the solutions of Dirac equation could be transformed to each other by
ψ(W )(p) = Sψ(D)(p). (3.27)
Thus, the Weyl solutions could be obtained through (3.27)
ψ+(W )(p) = Sψ(D)+ (p) = and ξ indicate that there may be two kinds of Lorentz scalars. They could be defined in
3.1. SPINOR NOTATIONS
the way
hpipji ≡ ¯ψ(D)− (pi) ψ(D)+ (pj) = η(pi)βη(pj)β = −hpjpii, (3.29) [pipj] ≡ ¯ψ(D)+ (pi) ψ(D)− (pj) = ξ(pi)β˙ξ(pj)β˙ = −[pjpi]. (3.30)
They are simply called the angle bracket for h· · · i and square bracket for [· · · ]. In the above, we have taken
|pii = η(pi)β, hpj| = η(pj)β, (3.31)
|pi] = ξ(pi)β˙, [pj| = ξ(pj)β˙. (3.32)
The spinors are also related to its null momentum by the identities
Pb ˙a ≡ (σ · p)˙ba = |pi[p|, (3.33) P˙ab ≡ (¯σ · p)˙ab = −|p]hp|. (3.34) Furthermore, some properties of these brackets could be derived by directly calculation:
[pipj] = hpjpii∗, (3.35)
hpipji[pjpi] = −2pi· pj. (3.36)
So far, some useful Lorentz scalars have been made by contracting the an upper dotted index with a lower dotted index spinor, or an upper undotted index with a lower undotted index such as in (3.29) and (3.30). As mentioned at the beginning of this section, we would like to use these Lorentz invariances as building blocks to build up scattering amplitudes. If we consider amplitudes include massless vector bosons, polarizations also require to be rewritten in the language of spinors. Polarization vectors with definite helicities for bosons can be represented as [12,13]
(k, q)µ+= hq|σµ|k]
√2hqki, (k, q)µ−= −[q|¯σµ|ki
√2[qk] . (3.37)
Here q is some chosen light–like momentum, called reference momentum.
3.2. CALCULATING COLOR–ORDERED 4–GLUON AMPLITUDE THROUGH