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Canonical Phase Space Constraints

Physical scope

Classical canonical phase-space form and first-class metric constraints of the flat tensile Polyakov string after conformal gauge fixing; no quantization claim.

Statement

For the adopted flat Polyakov model in conformal gauge $h_{a b} = diag (- 1, + 1)$ , the Lagrangian density is $L = \frac{T}{2} ({\dot{X}}^{2} - X^{' 2})$ , the canonical momentum is $Π^{μ} = T {\dot{X}}^{μ}$ , and the Hamiltonian density is $H = \frac{1}{2 T} Π^{2} + \frac{T}{2} X^{' 2}$ . The metric equation is equivalent to the constraints $D (σ) = Π \cdot X^{'} = 0$ and $G (σ) = Π^{2} + T^{2} X^{' 2} = 0$ . On $J = R / (2 π Z)$ for a closed string or $J = (0, π)$ for the open-string interior, use the canonical phase-space bracket specified in the Facts & Assumptions below. For smooth smearings $f, g$ that are periodic on the circle or compactly supported in the open-string interior, the brackets of $D [f] = \int_{J} f D d σ$ and $G [g] = \int_{J} g G d σ$ are ${D [f], D [g]} = D [f g^{'} - g f^{'}]$ , ${D [f], G [g]} = G [f g^{'} - g f^{'}]$ , and ${G [f], G [g]} = 4 T^{2} D [f g^{'} - g f^{'}]$ . Their closure as linear combinations of the constraints is the meaning of first-class here.

Statement references

Flat Bosonic String Model

String Tension Alpha Prime

Bosonic Polyakov Model

Conformal Gauge and Residual Symmetry

Polyakov Equations and Virasoro Constraints

Canonical Phase Space for the Flat String

Facts & Assumptions

Given: A smooth equal-time string configuration $X (σ)$ , momentum density $Π_{μ} (σ)$ , the flat-target metric $η$ , the adopted Polyakov action, and conformal gauge $h_{a b} = diag (- 1, + 1)$ .

[F1]

Flat Bosonic String Model: $X$ takes values in $R^{1, D - 1}$ with $η = diag (- 1, + 1, \dots, + 1)$ ; $τ, σ$ are dimensionless.

[F2]

String Tension Alpha Prime: $T = (2 π α^{'})^{- 1} > 0$ .

[A1]

Bosonic Polyakov Model: $S_{P} = - \frac{T}{2} \int \sqrt{- h} h^{a b} \partial_{a} X \cdot \partial_{b} X d^{2} σ$ .

[F3]

Conformal Gauge and Residual Symmetry: a local classical gauge choice permits $h_{a b} = e^{2 ϕ} diag (- 1, + 1)$ , and Weyl gauge sets $ϕ = 0$ .

[F4]

Polyakov Equations and Virasoro Constraints: the metric equation gives $T_{00} = \frac{1}{2} ({\dot{X}}^{2} + X^{' 2}) = 0$ and $T_{01} = \dot{X} \cdot X^{'} = 0$ .

[F5]

Canonical Phase Space for the Flat String: the canonical bracket is ${F, G} = \int_{J} (δ F / δ X^{μ} δ G / δ Π_{μ} - δ F / δ Π_{μ} δ G / δ X^{μ}) d σ$ , with periodic smearings on the circle and compactly supported smearings on the open-string interior.

Proof

1.1givenF1A1F3algebra

For the flat target and its mostly-plus metric [F1], substituting $h_{a b} = diag (- 1, + 1)$ into [A1] gives $L = - \frac{T}{2} (- {\dot{X}}^{2} + X^{' 2}) = \frac{T}{2} ({\dot{X}}^{2} - X^{' 2})$ .

1.2F5algebra

For smooth smearings $f, g$ of the boundary type in [F5], set $D [f] = \int_{J} f Π \cdot X^{'} d σ$ and $G [g] = \int_{J} g (Π^{2} + T^{2} X^{' 2}) d σ$ . Their variational derivatives are $δ D [f] / δ X = - (f Π)^{'}$ , $δ D [f] / δ Π = f X^{'}$ , $δ G [g] / δ X = - 2 T^{2} (g X^{'})^{'}$ , and $δ G [g] / δ Π = 2 g Π$ , by differentiating the integrands and integrating by parts with no boundary term.

2.1F2step 1.1algebra

Differentiating this density with respect to ${\dot{X}}_{μ}$ gives $Π^{μ} = \partial L / \partial {\dot{X}}_{μ} = T {\dot{X}}^{μ}$ . Since $T > 0$ , ${\dot{X}}^{μ} = Π^{μ} / T$ .

3.1step 1.1step 2.1algebra

The Legendre transform is $H = Π_{μ} {\dot{X}}^{μ} - L = T {\dot{X}}^{2} - \frac{T}{2} ({\dot{X}}^{2} - X^{' 2}) = \frac{T}{2} ({\dot{X}}^{2} + X^{' 2})$ . Substituting step 2.1 yields $H = \frac{1}{2 T} Π^{2} + \frac{T}{2} X^{' 2}$ .

3.2F2F4step 2.1algebra

Multiplying the two metric constraints in [F4] by $T$ and $T^{2}$ , respectively, and using step 2.1 gives $D = Π \cdot X^{'} = T \dot{X} \cdot X^{'} = 0$ and $G = Π^{2} + T^{2} X^{' 2} = T^{2} ({\dot{X}}^{2} + X^{' 2}) = 0$ . Conversely, because $T > 0$ , these canonical constraints imply both metric constraints.

4.1F5step 1.2algebra∎

Substituting the derivatives from step 1.2 into the bracket, the first constraint bracket is ${D [f], D [g]} = \int_{J} (- (f Π)^{'} \cdot g X^{'} + f X^{'} \cdot (g Π)^{'}) d σ = \int_{J} (f g^{'} - g f^{'}) Π \cdot X^{'} d σ,$ because the $Π^{'} \cdot X^{'}$ terms cancel. For the mixed bracket, expand the derivatives: $- 2 g Π \cdot (f Π)^{'} = - 2 g f^{'} Π^{2} - f g (Π^{2})^{'}, 2 T^{2} f X^{'} \cdot (g X^{'})^{'} = 2 T^{2} f g^{'} X^{' 2} + T^{2} f g (X^{' 2})^{'} .$ Integration by parts on the total-derivative terms, with no endpoint term by [F5], makes their sum $(f g^{'} - g f^{'}) (Π^{2} + T^{2} X^{' 2})$ , hence ${D [f], G [g]} = G [f g^{'} - g f^{'}]$ . Finally, ${G [f], G [g]} = 4 T^{2} \int_{J} (- g Π \cdot (f X^{'})^{'} + f Π \cdot (g X^{'})^{'}) d σ = 4 T^{2} D [f g^{'} - g f^{'}],$ since the terms proportional to $f g Π \cdot X^{''}$ cancel. Thus all three brackets are smearings of the constraints, which is first-class closure by definition here.

Canonical Phase Space Constraints

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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