A Positive Formalism for Quantum Theory in the General Boundary Formulation

Abstract

We introduce a new “positive formalism” for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory.

Appendix: GBF Axioms in the Amplitude Formalism

1.1 A.1 Spacetime System

We recall aspects of the way spacetime is encoded in the GBF abstractly through a spacetime system [2, 5]. The latter consists of:

• A collection of oriented topological manifolds of dimension d with boundary and possibly with additional structure. These are called regions.

• A collection of oriented topological manifolds of dimension d−1 without boundary and possibly with additional structure. These are called hypersurfaces.

These collections satisfy additional properties:

• Regions and hypersurfaces may only have a finite number of connected components.

• The boundary of a region is a hypersurface.

• Every connected component of a region is a region and every connected component of a hypersurface is a hypersurface.

• There is a notion of decomposition of a hypersurface which consists in presenting the hypersurface as a disjoint union of other hypersurfaces.

• There is a notion of gluing of regions which consists in presenting a region as the union of regions such that the interiors are disjoint and the intersection of the original regions is a hypersurface.

There is also a modified notion of region, called a slice region.⁸ This is really a hypersurface Σ that is treated as if it was a region, denoted $\hat{\varSigma}$ with boundary $\varSigma\cup \overline{\varSigma}$.

1.2 A.2 Core Axioms

The core axioms of the GBF are presented here in the form given in [5]. In contrast to earlier version, this permits the inclusion of fermionic degrees of freedom. The latter come with a $\mathbb{Z} _{2}$-grading, the fermionic or f-grading.

(T1):

Associated to each hypersurface Σ is a complex separable f-graded Krein space $\mathcal{H}_{\varSigma}$. We denote its indefinite inner product by 〈⋅,⋅〉 _Σ .

(T1b):

Associated to each hypersurface Σ is a conjugate linear adapted f-graded isometry $\iota_{\varSigma}:\mathcal{H}_{\varSigma}\to\mathcal{H} _{\overline{\varSigma}}$. This map is an involution in the sense that $\iota_{\overline{\varSigma}}\circ\iota_{\varSigma}$ is the identity on $\mathcal{H}_{\varSigma}$.

(T2):

Suppose the hypersurface Σ decomposes into a disjoint union of hypersurfaces Σ=Σ ₁∪⋯∪Σ _n . Then, there is an isometric isomorphism of Krein spaces $\tau_{\varSigma_{1},\dots,\varSigma_{n};\varSigma}:\mathcal{H}_{\varSigma _{1}}\hat{\otimes} \cdots\hat{\otimes}\mathcal{H}_{\varSigma_{n}}\to\mathcal{H}_{\varSigma}$. The maps τ satisfy obvious associativity conditions. Moreover, in the case n=2 the map $\tau_{\varSigma_{2},\varSigma_{1};\varSigma}^{-1}\circ\tau_{\varSigma _{1},\varSigma_{2};\varSigma}:\mathcal{H}_{\varSigma_{1}}\hat{\otimes}\mathcal {H}_{\varSigma_{2}}\to\mathcal{H} _{\varSigma_{2}}\hat{\otimes}\mathcal{H}_{\varSigma_{1}}$ is the f-graded transposition,

$$ \psi_1\otimes\psi_2\mapsto(-1)^{|\psi_1|\cdot|\psi_2|}\psi _2\otimes\psi_1 . $$

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(T2b):

Orientation change and decomposition are compatible in an f-graded sense. That is, for a disjoint decomposition of hypersurfaces Σ=Σ ₁∪Σ ₂ we have

$$ \tau_{\overline{\varSigma}_1,\overline{\varSigma}_2;\overline{\varSigma}} \bigl(\iota_{\varSigma_1}(\psi_1)\otimes \iota_{\varSigma_2}(\psi _2) \bigr) =(-1)^{|\psi_1|\cdot|\psi_2|} \iota_\varSigma \bigl(\tau _{\varSigma_1,\varSigma_2;\varSigma}(\psi_1\otimes \psi_2) \bigr) . $$

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(T4):

Associated to each region M is a f-graded linear map from a dense subspace $\mathcal{H}_{\partial M}^{\circ}$ of $\mathcal{H}_{\partial M}$ to the complex numbers, $\rho_{M}:\mathcal{H}_{\partial M}^{\circ}\to\mathbb{C}$. Here ∂M denotes the boundary of M with the induced orientation. This is called the amplitude map.

(T3x):

Let Σ be a hypersurface. The boundary $\partial \hat{\varSigma}$ of the associated empty region $\hat{\varSigma}$ decomposes into the disjoint union $\partial\hat{\varSigma}=\overline {\varSigma}\cup\varSigma'$, where Σ′ denotes a second copy of Σ. Then, $\rho_{\hat{\varSigma}}$ is well defined on $\tau _{\overline{\varSigma},\varSigma';\partial\hat{\varSigma}}(\mathcal {H}_{\overline {\varSigma}}\otimes\mathcal{H}_{\varSigma'})\subseteq\mathcal {H}_{\partial\hat{\varSigma }}$. Moreover, $\rho_{\hat{\varSigma}}\circ\tau_{\overline{\varSigma },\varSigma';\partial\hat{\varSigma}}$ restricts to a bilinear pairing $(\cdot,\cdot)_{\varSigma}:\mathcal{H}_{\overline{\varSigma}}\times \mathcal{H}_{\varSigma '}\to\mathbb{C}$ such that 〈⋅,⋅〉 _Σ =(ι _Σ (⋅),⋅) _Σ .

(T5a):

Let M ₁ and M ₂ be regions and M:=M ₁∪M ₂ be their disjoint union. Then ∂M=∂M ₁∪∂M ₂ is also a disjoint union and $\tau_{\partial M_{1},\partial M_{2};\partial M}(\mathcal{H}_{\partial M_{1}}^{\circ}\otimes\mathcal {H}_{\partial M_{2}}^{\circ})\subseteq\mathcal{H}_{\partial M}^{\circ}$. Moreover, for all $\psi _{1}\in\mathcal{H} _{\partial M_{1}}^{\circ}$ and $\psi_{2}\in\mathcal{H}_{\partial M_{2}}^{\circ}$,

$$ \rho_{M} \bigl(\tau_{\partial M_1,\partial M_2;\partial M}(\psi _1\otimes \psi_2) \bigr)= \rho_{M_1}(\psi_1) \rho_{M_2}(\psi_2) . $$

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(T5b):

Let M be a region with its boundary decomposing as a disjoint union $\partial M=\varSigma_{1}\cup\varSigma\cup\overline{\varSigma '}$, where Σ′ is a copy of Σ. Let M ₁ denote the gluing of M with itself along $\varSigma,\overline{\varSigma'}$ and suppose that M ₁ is a region. Note ∂M ₁=Σ ₁. Then, $\tau_{\varSigma_{1},\varSigma,\overline{\varSigma'};\partial M}(\psi\otimes \xi\otimes\iota_{\varSigma}(\xi))\in\mathcal{H}_{\partial M}^{\circ}$ for all $\psi\in\mathcal{H}_{\partial M_{1}}^{\circ}$ and $\xi\in\mathcal {H}_{\varSigma}$. Moreover, for any orthonormal basis {ζ _i } _i∈I of $\mathcal{H} _{\varSigma}$ in $\mathcal{H}^{\circ}_{\varSigma}$, we have for all $\psi \in\mathcal{H} _{\partial M_{1}}^{\circ}$,

$$ \rho_{M_1}(\psi)\cdot c\bigl(M;\varSigma,\overline{ \varSigma'}\bigr) =\sum_{i\in I}(-1)^{[\zeta_i]} \rho_M \bigl(\tau_{\varSigma _1,\varSigma,\overline{\varSigma'};\partial M}\bigl(\psi\otimes\zeta _i\otimes \iota_\varSigma(\zeta_i)\bigr) \bigr), $$

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where $c(M;\varSigma,\overline{\varSigma'})\in\mathbb{C}\setminus\{0\}$ is called the gluing anomaly factor and depends only on the geometric data.

1.3 A.3 Observable Axioms

Observables were introduced into the GBF and axiomatized in [8]. We extend here the axiomatization in the form presented in [9] from the purely bosonic case to the general case.⁹

(O1):

Associated to each spacetime region M is a real vector space $\mathcal{O}_{M}$ of linear maps $\mathcal{H}_{\partial M}^{\circ}\to\mathbb{C}$, called observable maps. In particular, $\rho_{M}\in\mathcal{O}_{M}$.

(O2a):

Let M ₁ and M ₂ be regions and M=M ₁∪M ₂ be their disjoint union. Then, there is an injective bilinear map $\diamond :\mathcal{O}_{M_{1}}\times\mathcal{O}_{M_{2}}\hookrightarrow\mathcal {O}_{M}$ such that for all $\rho ^{O_{1}}_{M_{1}}\in\mathcal{O}_{M_{1}}$ and $\rho^{O_{2}}_{M_{2}}\in\mathcal {O}_{M_{2}}$ and $\psi_{1}\in\mathcal{H}_{\partial M_{1}}^{\circ}$ and $\psi_{2}\in \mathcal{H}_{\partial M_{2}}^{\circ}$,

$$ A^{O_1}_{M_1}\diamond A^{O_2}_{M_2}( \psi_1\otimes\psi_2) = A^{O_{1}}_{M_1}( \psi_1) A^{O_{2}}_{M_2}(\psi_2) . $$

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This operation is required to be associative in the obvious way.

(O2b):

Let M be a region with its boundary decomposing as a disjoint union $\partial M=\varSigma_{1}\cup\varSigma\cup\overline{\varSigma '}$ and M ₁ given as in (T5b). Then, there is a linear map $\diamond _{\varSigma}: \mathcal{O}_{M}\to~\mathcal{O}_{M_{1}}$ such that for all $A^{O}_{M}\in\mathcal{O} _{M}$ and any orthonormal basis {ξ _i } _i∈I of $\mathcal {H}_{\varSigma}$ in $\mathcal{H}_{\varSigma}^{\circ}$ and for all $\psi\in\mathcal {H}_{\partial M_{1}}^{\circ}$,

$$ \diamond_{\varSigma}\bigl(A^O_M\bigr) (\psi)\cdot c \bigl(M;\varSigma,\overline{\varSigma'}\bigr) =\sum _{i\in I} (-1)^{[\xi_i]} A^O_M\bigl( \psi\otimes\xi_i\otimes \iota _\varSigma(\xi_i) \bigr) . $$

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This operation is required to commute with itself and with (O2a) in the obvious way.