Probabilistic Time

Abstract

The concept of time emerges as an ordering structure in a classical statistical ensemble. Probability distributions p _τ (t) at a given time t obtain by integrating out the past and future. We discuss all-time probability distributions that realize a unitary time evolution as described by rotations of the real wave function $q_{\tau}(t)=\pm \sqrt{p_{\tau}(t)}$. We establish a map to quantum physics and the Schrödinger equation. Suitable classical observables are mapped to quantum operators. The non-commutativity of the operator product is traced back to the incomplete statistics of the local-time subsystem. Our investigation of classical statistics is based on two-level observables that take the values one or zero. Then the wave functions can be mapped to elements of a Grassmann algebra. Quantum field theories for fermions arise naturally from our formulation of probabilistic time.

Appendices

Appendix A: Evolution Law with Transition Matrix

In this appendix we demonstrate that an evolution law with a transition matrix (41) does, in general, not account for periodic probabilities. The main reason is the lack of time reversal symmetry. This does, however, not exclude particular evolution laws based on a class of transition matrices for which time reflection symmetry can be realized.

The evolution law (41) entails a composition property

(191)

This can be “integrated”, and we obtain for t ₃≥t ₂≥t ₁≥t ₀ the matrix multiplication

$$ W(t_3,t_1)=W(t_3,t_2)W(t_2,t_1), $$

(192)

with

$$ W(t,t)=1. $$

(193)

We can then express p _τ (t) in terms of an “initial” probability distribution p _τ (t ₀),

$$ p_\tau(t)=\sum_\rho W_{\tau\rho}(t,t_0)p_\rho(t_0). $$

(194)

We may interpret the probabilities p _τ (t) as components of a vector in $\mathbb{ R}^{\mathcal{N}}$. The basis vectors of $\mathbb{R}^{\mathcal{N}}$ are the “classical states” ρ for which $p^{(\rho)}_{\tau}=\delta^{\rho}_{\tau}$. The transition matrices act then as operators in this space.

In turn, each classical state is a sequence of occupation numbers for B bits, and we can characterize W by its operation on the bits. Consider the example of two bits $(B=2, \mathcal{N}=4)$, where ρ=[(1,1),(1,0),(0,1),(0,0)]. We can compose W as a sum of “elementary bit operators” B ^(v),

$$ W=\sum_v\kappa_vB^{(v)},\quad v=1 \cdots\bigl(\mathcal{N}^2-1\bigr). $$

(195)

A typical bit operator is the annihilation operator $B^{(1)}_{-}$ for bit 1 which changes n ₁=1 to n ₁=0 and yields zero when applied to a classical state for which n ₁=0. It maps (0,0)→0,(0,1)→0,(1,0)→(0,0),(1,1)→(0,1) and corresponds to

$$ B^{(1)}_-=\left (\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0,&0,&0,&0\\0,&0,&0,&0\\1,&0,&0,&0\\0,&1,&0,&0 \end{array} \right ) ,\qquad B^{(1)}_+= \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0,&0,&1,&0\\0,&0,&0,&1\\0,&0,&0,&0\\0,&0,&0,&0 \end{array} \right ). $$

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The hermitean conjugate $(B^{(1)}_{-})^{\dagger}=B^{(1)}_{+}$ is the creation operator for bit 1. Other operators are the switch between two bits (0,1)↔(1,0), the diagonal number operators N ₁=diag(1,1,0,0),N ₂=diag(1,0,1,0) or their product N ₁ N ₂=diag(1,0,0,0). The coefficients κ _v have to obey restrictions in order to obey the relations (42), (43). An example is $W=B^{(1)}_{-}+B^{(1)}_{+}$, with W ²=1.

The simple periodic oscillations of a local two-state probability (B=1), as given by Eq. (33), cannot be expressed by transition matrices obeying Eqs. (42), (43). Indeed, the most general form of W _τρ for B=1 reads

(197)

If we characterize W ₁,W ₂ by p ₁,p ₂, the product W=W ₁ W ₂=W ₂ W ₁ has the form (197) with

$$ p=p_1(1-p_2)+p_2(1-p_1). $$

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Thus W ₁ does not admit an inverse except for p ₁=0,1, since p differs from zero for arbitrary p ₂. An inverse of W would be needed if we want to represent the time reflected evolution of the local probabilities—we note that Eq. (33) is invariant under t→−t.

If we want to realize a continuous evolution in the limit ϵ→0 we should have for W(t,t−ϵ) an infinitesimal form p=Aϵ,A≠0. The time evolution obeys then the differential equation

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For A>0 this corresponds to an irreversible approach to the fixed point at p ₁=p ₀=1/2 rather than to an oscillation. Similar problems persists for a higher number of species B and we conclude that an evolution law involving transition matrices is often not suitable for a description of oscillating local probabilities. There are exceptional cases where a linear law of evolution can describe periodic probabilities and we will mention a few of them in this paper.

Appendix B: Evolution Operator and Quantum Formalism

2.1 B.1 Grassmann Evolution Operator

In the Grassmann representation the close correspondence of the “unitary evolution law” given by Eqs. (44), (49) or (48) to the time evolution in quantum mechanics can be made apparent. For this purpose we employ operators $\mathcal{U}(t,t_{0})$ acting in the Grassmann algebra. They describe the time evolution of a Grassmann element g(t)=∑ _τ c _τ (t)g _τ according to

$$ g(t)=\mathcal{U}(t,t_0)g(t_0) ,\qquad \tilde{g}(t)= \tilde{\mathcal{U}}^T(t,t_0)\tilde{g}(t_0). $$

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Here we define $\tilde{\mathcal{U}}^{T}(t,t_{0})$ such that for arbitrary elements $\tilde{g},f$ of the Grassmann algebra one has

$$ \int \mathcal{D}\psi\tilde{\mathcal{U}}^T\tilde{g} f=\int \mathcal{D}\psi\tilde{g} \tilde{\mathcal{U}}f. $$

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The associated time evolution of p _τ (t) obtains for a given evolution operator $\mathcal{U}(t,t_{0})$ as

(202)

We may express the linear operators $\mathcal{U},\tilde {\mathcal {U}}^{T}$ as matrix multiplications

(203)

with

$$ \tilde{U}^T_{\tau\rho}=\tilde{U}_{\rho\tau}, $$

(204)

such that

(205)

From ∑ _τ p _τ =1 we infer

$$ \sum_{\rho,\sigma} (\tilde{U}U)_{\rho\sigma}c_\rho^*c_\sigma = 1, $$

(206)

which can hold for arbitrary c _τ obeying ∑ _τ |c _τ |²=1 only if $(\tilde{U}U)_{\rho\sigma}=\delta_{\rho\sigma}$. Thus the matrix $\tilde{U}$ is the inverse of U. Furthermore, the condition that p _τ (t) is real is obeyed by Eq. (205) if $\tilde{U}^{T}= U^{*}$. Then p _τ (t)≥0 follows automatically

$$ p_\tau(t)=\bigg\vert\sum_\sigma U_{\tau\sigma}(t,t_0)c_\sigma(t_0) \bigg\vert^2. $$

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We conclude that U is a unitary matrix, UU ^†=1, $\tilde{U}=U^{\dagger}$. Already at this stage we recognize the unitary time evolution operator of quantum mechanics. The Grassmann operator $\mathcal{U}$ plays a role similar to the unitary evolution operator in quantum mechanics,

$$ \tilde {\mathcal {U}}(t,t_0) \mathcal{U}(t,t_0)= 1_\mathcal{G}. $$

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For the case of real c _τ =q _τ we have to require that q _τ =∑ _ρ U _τρ q _ρ is real. This restricts the unitary transformations to the subgroup of real matrices U. Those are the rotations $SO(\mathcal{N})$, or U=R,R ^T R=1, and we recover Eq. (48) if we consider the g _τ as a fixed basis of the Grassmann algebra.

We note that the Grassmann algebra admits an involution

$$ g_\tau \leftrightarrow \tilde{g}_\tau ,\qquad c_\tau \leftrightarrow c^*_\tau ,\qquad g \leftrightarrow \tilde{g}. $$

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This can be used in order to define the notion of a complex conjugation in the Grassmann algebra

$$ (c_\tau g_\tau)^*=c_\tau^* \tilde{g}_\tau, $$

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which is a nontrivial operation even for real c _τ =q _τ . In terms of this complex structure we can regard $\tilde {\mathcal {U}}$ as the hermitean conjugate of $\mathcal{U}$.

2.2 B.2 Schrödinger Equation

The analogy to quantum mechanics is apparent if we interpret the Grassmann-valued wave function g as a vector with components

$$ \varphi_\tau(t)=\mathcal{P}_\tau g(t)=c_\tau(t)g_\tau. $$

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The complex conjugate wave function $g^{*}=\tilde{g}$ has components

$$ \tilde{\varphi}_\tau(t)=\mathcal{P}_\tau\tilde{g}(t)=c^*_\tau(t)\tilde{g}_\tau= \varphi^*_\tau(t). $$

(212)

In terms of the components we can write

$$ p_\tau(t)=\int \mathcal {D}\psi\tilde{\varphi}_\tau(t) \varphi_\tau(t). $$

(213)

Expectation values of observables find an expression similar to quantum mechanics,

$$ \big\langle A(t)\big\rangle =\int \mathcal {D}\psi\sum_\tau\tilde{\varphi}_\tau(t)\mathcal{A}\varphi_\tau(t), $$

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with $\mathcal{A}$ acting as a diagonal operator $\mathcal{A}\varphi_{\tau}=A_{\tau}\varphi_{\tau}$.

In this picture the time evolution can be written as the usual unitary evolution of the wave function

(215)

which yields explicitly

(216)

Here we have introduced the quantum mechanical “density matrix”

$$ \rho_{\tau\sigma}=\int \mathcal {D}\psi\tilde{\varphi}_\tau \varphi_\sigma ,\quad \rho=\rho^\dagger ,\quad \text{tr}\,\rho=1, $$

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with real diagonal elements ρ _ττ =p _τ obeying 0≤ρ _ττ ≤1.

The quantum mechanical formalism can also be presented in the standard way by omitting the basis vectors g _τ and defining complex vectors φ and φ ^∗ with components $c_{\tau},c^{*}_{\tau}$, with associated density matrix $\rho_{\tau\sigma}=c_{\tau}c^{*}_{\sigma}$. The classical observables can be represented as diagonal operators $\hat{A},(\hat{A})_{\tau\sigma}=A_{\tau}\delta_{\tau\rho}$, and we can define the usual scalar product (with |φ〉=φ,〈φ|=φ ^∗), such that

$$ \langle A\rangle =\langle \varphi|\hat{A}|\varphi\rangle =\text{tr}(\rho\hat{A}). $$

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Even though we describe a classical statistical ensemble, the unitary time evolution (44), (48) can be fully described with the formalism of quantum mechanics. Of course, at this stage the observables all commute and are all represented by diagonal operators. Furthermore, the wave function φ remains real if we use the real Grassmann algebra with $c_{\tau}=c^{*}_{\tau}=q_{\tau}$.

The time evolution of the Grassmann wave function g(t) obeys a differential evolution law

$$ i\partial_tg(t)=\mathcal{H}(t)g(t) $$

(219)

with Hamilton operator $\mathcal{H}$ defined by Eq. (84). In terms of the components of the wave function Eq. (219) becomes

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This shows explicitly that the Schrödinger equation can be obtained from an appropriate evolution law for classical probabilities.