Spacetime quantum actions

We propose a formulation of quantum mechanics in an extended Fock space in which a tensor product structure is applied to time. Subspaces of histories consistent with the dynamics of a particular theory are defined by a direct quantum generalization of the corresponding classical action. The diagonalization of such quantum actions enables us to recover the predictions of conventional quantum mechanics and reveals an extended unitary equivalence between all physical theories. Quantum correlations and coherent effects across time and between distinct theories acquire a rigorous meaning, which is encoded in the rich temporal structure of physical states. Connections with modern relativistic schemes and the path integral formulation also emerge.

Figure 1

Representation of two classical (distinguishable) particles moving in flat spacetime whose trajectories can be parametrized as $(t,q_a(t),q_b(t))$ (top left). Conventional QM describes this situation by employing a basis of product states $|\mathit{q}⟩=|q_a⟩\otimes |q_b⟩$ which represent the positions at a given time in the Hilbert space $\mathfrak{H}$. Instead, in $\mathcal{H}$, the whole paths are represented by $|\mathit{q}(t)⟩=|q_a(t)⟩\otimes |q_b(t)⟩\propto {\otimes }_j|q_{a{t}_j}⟩\otimes |q_{b{t}_j}⟩$ [Eq. (7)], where $|q_i(t)⟩\propto {\otimes }_j|q_{i{t}_j}⟩$ (top right), which establishes a completely symmetric application of the tensor product to spatial and temporal degrees of freedom. Moreover, classical time evolution $\mathit{q}(t)\to \mathit{q}(t+\mathrm{\Delta }t)$ can be seen from a passive point of view as a displacement $t\to t-\mathrm{\Delta }t$ of the whole manifold. In our formulation, quantum time evolution emerges from $e^{i{\mathcal{P}}_t(-\mathrm{\Delta }t)}|\mathit{q}(t)⟩=|\mathit{q}(t+\mathrm{\Delta }t)⟩$. The symmetry between space and time is further depicted on the bottom panel with a different example: The tensor product in space of a conventional quantum field theory is extended here to spacetime.

Figure 2

On the left, the two descriptions of the single particle: the conventional one in the Hilbert space $\mathfrak{H}$ (top panel) and the generalized description in spacetime in the Hilbert $\mathcal{K}$ (bottom panel). On the right, the second quantization of the previous schemes. The second quantization of $\mathfrak{H}$ leads to a field theory in a conventional Hilbert space ${\mathfrak{H}}^F$ which is isomorphic to a tensor product in space of copies of $\mathfrak{H}$, i.e., ${\mathfrak{H}}^F\approx {\otimes }_q{\mathfrak{H}}_q$ (top right panel). The second quantization of $\mathcal{K}$ leads instead to an extended space $\mathcal{H}\approx {\otimes }_t{\mathfrak{H}}_t^F={\otimes }_{t,q}{\mathfrak{H}}_{tq}$ where the tensor product structure is applied to both space and time and it is possible to represent field configurations in spacetime (bottom right panel). The description of the field in this extended Hilbert space can be immediately obtained by applying the formalism presented in this work to this particular case.

Figure 3

Under time translations through $\mathrm{\Delta }t/\epsilon$ steps, the operator ${\tilde{\mathit{A}}}_{t_j}$ is displaced to site $t_{j^{\prime }}=t_j+\mathrm{\Delta }t$ while evolving an amount $\mathrm{\Delta }t$ (left panel). Through insertion of operators at different times and translations back to the Hilbert at $t=0$, multiple-time correlation functions are obtained (right panel).