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e tigfil theory field string sed ϕ 0a ie0 time at =0 G d hc obeys which . (1.1) (1.2) UK , then generalise to string field theory. The essential ingredient is the set of gluing properties. For the scalar field propagator these are
iG0(x2,t2; x1,t1) t2 >t>t1 D ∂ −iG0(x2,t2; x1,t1) t1 >t>t2 d y G0(x2,t2; y,t) −2 G0(y,t; x1,t1)= Z ∂t iGI (x2,t2; x1,t1) t>t1,t2 −iGI (x2,t2; x1,t1) t
2 The string field vacuum state
The vacuum state functional is the generator of vacuum expectation values. For example, the open string three field expectation value would be represented by, to leading order in
the coupling, the diagram
X
¿
X
¾
X
½
= ¼ ØiÑe (2.1) Such expectation values can be used to reconstruct the vacuum state functional. We will demonstrate this for quantum field theory and then generalise to strings. Since the string field interaction is in various guises cubic we will consider a non-local scalar φ3 theory with interaction Hamiltonian density
3 λ D d xj dtj W (t, x; x1,t1, x2,t2, x3,t3)φ(x1,t1)φ(x2,t2)φ(x3,t3) (2.2) 3! Z jY=1 written in terms of some kernel W . Replacing this kernel with a product of delta functions 3 δ(t − ti) recovers the local φ Hamiltonian density. The technique we are about to describe applies to both local and non-local interactions, as demonstrated in [6] for local φ4 theory. To begin, consider the free field vacuum functional, which must yield the equal time propagator as a vacuum expectation value,
~G(x, 0; y, 0) = h 0 |φ(x, 0)φ(y, 0)| 0 i = Dφ φ(x)φ(y)Ψ2[φ] Z 0 which implies that the free field vacuum is the exponent of the inverse of the equal time propagator. This inverse is
2 −1 ∂ ′ G0(x, 0; y0) = 4 ′ G0(x,t; yt) (2.3) ∂t∂t ′ t=t =0
2 as we proved in [6], so the free field vacuum is
• • free 1 D Ψ [φ] = exp − d (x, y)φ(x)G0(x, 0; y, 0)φ(y) , (2.4) 0 4~ Z as is easily checked. The bullet is a time derivative with factor −2, as in (1.3). Now introduce the cubic interaction. Our arguments are based in the Schr¨odinger representation, and treats all spatial arguments in the same way as covariant methods. Therefore we may choose the interaction to be local or non-local in space without affecting the results, as such we will write the Hamiltonian as
3 λ dtj W (t; t1,t2,t3)φ(t1)φ(t2)φ(t3) (2.5) 3! Z jY=1 to keep notation compact, suppressing the spatial dependencies whenever possible. Accord- ingly we will abbreviate our representation of the free propagator to
Gx(0; tj ) := G0(x, 0; xj ,tj ),
D (2.6) φxGx(0; tj ) := d x φ(x)G0(x, 0; xj ,tj ). Z To clarify our arguments we also assume that the interaction kernel is symmetric in its indices. The following applies when this is not the case, the only difference being the ap- pearance of sums of terms corresponding to inequivalent ways of attaching propagators. We will now describe how to identify higher order terms in the vacuum functional by expanding in orders of λ and ~ with a recursive process using vacuum expectation values. We begin with the lowest order expectation value, which is the tree level three field expectation value of order λ~2. A covariant field theory calculation implies that this is
∞ ∞ 3 2 h 0 |φ(x, 0)φ(y, 0)φ(z, 0)| 0 i = −iλ~ dt dtj W (t; t1,t2,t3) Z Z (2.7) −∞ −∞ jY=1
×Gx(0; t1)Gy(0; t2)Gz(0; t3)
Alternatively, we can abandon the explicit kernel W and think of this as some Feynman diagram which ties together three legs ending at x0 = 0, in some way (the usual Feynman
diagram showing three propagators meeting at a point vertex is not suitable):
¾
Ø Ø Ø
½ ¾ ¿
ih
¼
Ý Þ Ü (2.8)
Introducing an unknown diagram Γ into the vacuum,
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