Mind–Body Interaction and Modern Physics

Abstract

The idea that mind and body are distinct entities that interact is often claimed to be incompatible with physics. The aim of this paper is to disprove this claim. To this end, we construct a broad mathematical framework that describes theories with mind–body interaction (MBI) as an extension of current physical theories. We employ histories theory, i.e., a formulation of physical theories in which a physical system is described in terms of (i) a set of propositions about possible evolutions of the system and (ii) a probability assignment to such propositions. The notion of dynamics is incorporated into the probability rule. As this formulation emphasises logical and probabilistic concepts, it is ontologically neutral. It can be used to describe mental ‘degrees of freedom’ in addition to physical ones. This results into a mathematical framework for psycho-physical interaction ($\Psi \Phi$I formalism). Interestingly, a class of $\Psi \Phi$I theories turns out to be compatible with energy conservation.

Appendices

Appendix 1: The Structure of Histories Theory

In this Appendix, we briefly summarize some features of histories theory, both classical and quantum.

1.1 Classical Histories Theory

Consider a system described by a sample space $\Gamma$ that is a differentiable manifold. Single-time propositions corresponds to measurable subsets C of $\Gamma$. Let $\mathcal{T} = \{t_1, t_2, \ldots , t_n\}$ be a discrete time-set for this system with $t_1< t_2< \ldots < t_n$. In general, $\mathcal{T}$ may be any partially ordered set, but here we restrict our considerations to the simplest case.

The path space $\Pi$ consists of all maps $\gamma :\mathcal{T} \rightarrow \Gamma$. The set of history propositions $\mathcal{V}$ consists of all measurable subsets of $\Pi$. Hence, we can write a history proposition for this system as

$$\alpha = (C_{t_{1}}, C_{t_{2}}, \ldots , C_{t_{n}}),$$

(8)

where $C_{t_i}$ is a subset of $\Gamma$ that corresponds to the proposition that the system was found in $C_{t_i}$ at time $t_i$. Logical operations are standardly defined in terms of set-theoretic operations between subsets of $\Pi$.

A probability functional assigns a probability $\text{ Prob }(\alpha )$ to each history proposition $\alpha \in \mathcal{V}$, as

$$\begin{aligned} \text{ Prob }(\alpha ) = \int dx_1 \ldots dx_n \chi _{C_{t_1}}(x_1) \chi _{C_{t_2}}(x_2) \ldots \chi _{C_{t_n}}(x_n) p(x_1, t_1; x_2, t_2; \ldots , x_n, t_n), \end{aligned}$$

(9)

where $p(x_1, t_1; x_2, t_2; \ldots , x_n, t_n)$ is a probability measure on $\Pi$, and $\chi _C$ the characteristic function of the set $C \subset \Gamma$.

The theory of stochastic processes is an example of a classical history theory. Of particular interest are Markovian processes, which describe probabilistic systems without memory. The associate probability measures are of the form

$$\begin{aligned} p(x_1, t_1; x_2, t_2; \ldots , x_n, t_n) = \rho _{t_1}(x_1) g(x_2, t_2|x_1, t_1) g(x_3, t_3|x_2, t_2) \ldots g(x_n, t_n|x_{n-1}, t_{n-1}) , \end{aligned}$$

(10)

where $\rho _{t_1}$ is a probability density on $\Gamma$ at the initial moment of time, and $g(x_2, t_2|x_1, t_1) > 0$ is the transition matrix between times $t_1$ and $t_2$. The transition matrix is stochastic, i.e., it satisfies $\int dx_2 g(x_2, t_2|x_1, t_1) = 1$.

Deterministic processes (without memory) correspond to the degenerate case where the transition matrix is a delta function, i.e.,

$$\begin{aligned} g(x_2, t_2|x_1, t_1) = \delta [x_2 - f_{t_2, t_1}(x_1)], \end{aligned}$$

(11)

where $f_{t,t'}$ is a diffeomorphism on $\Gamma$ indexed by t and $t'$. For classical mechanics and field theory, see, Ref. [32].

1.2 Quantum Histories Theory

In Copenhagen quantum mechanics, propositions correspond to measurement outcomes. Ideal measurements are described in terms of projection-valued measures (PVMs) on a Hilbert space $\mathcal{H}$, i.e., a family of projectors ${\hat{P}}_{a}$ indexed by a, that satisfy the following properties: (i) mutual exclusion, ${\hat{P}}_a {\hat{P}}_b = \delta _{ab} {\hat{P}}_a$, and (ii) exhaustion $\sum _a {\hat{P}}_a = {\hat{I}}$.

A sequence of measurements at times $t_1, t_2, \ldots , t_n$ corresponds to a sequence of PVMs ${\hat{P}}_{a_i, t_i}$ indexed by the time-parameter. A history of measurement outcomes is a sequence of projectors

$$\begin{aligned} \alpha := ({\hat{P}}_{a_1, t_1}, {\hat{P}}_{a_2, t_2}, \ldots , {\hat{P}}_{a_n, t_n}). \end{aligned}$$

(12)

Suppose that the system is prepared at the state ${\hat{\rho }}_0$ at $t = 0$, and that its Hamiltonian is ${\hat{H}}$. The probability associated to $\alpha$ is

$$\begin{aligned} \text{ Prob }(\alpha ) = Tr\left( {\hat{C}}_{\alpha } {\hat{\rho }}_0 {\hat{C}}_{\alpha }^{\dagger }\right) , \end{aligned}$$

(13)

where

$$\begin{aligned} {\hat{C}}_{\alpha } = {\hat{P}}_{a_n, t_n}(t_n) \ldots {\hat{P}}_{a_2, t_2} {\hat{P}}_{a_1, t_1}(t_1) \end{aligned}$$

(14)

is an operator associated to the history $\alpha$, defined in terms of the Heisenberg-picture projectors ${\hat{P}}_{a_i, t_i}(t_i) = e^{i{\hat{H}}t_i} {\hat{P}}_{a_i, t_i}e^{-i{\hat{H}}t_i}$.

The consistent/decoherent histories interpretation of quantum mechanics starts with the observation that the sequence $({\hat{P}}_{a_1, t_1}, {\hat{P}}_{a_2, t_2}, \ldots , {\hat{P}}_{a_n, t_n})$ can be interpreted as referring to propositions about properties of a physical system, and not only to measurement outcomes. Then, it is possible to define logical operations, such as $\text{ AND }$, $\text{ OR }$, $\text{ NOT }$ and so on, between those history propositions, and to define a set $\mathcal{V}$ if history propositions that is closed under those operations.

The set of history propositions is constructed in terms of the history Hilbert space $\mathcal{K}_{his}$, defined as the tensor product of single time Hilbert spaces,

$$\begin{aligned} \mathcal{K}_{his} = \otimes _{t \in \mathcal{T}} \mathcal{H}_t, \end{aligned}$$

(15)

where t is an element of the time-set $\mathcal{T}$, and $\mathcal{H}$ is a copy of the single-time Hilbert space indexed by t. The history (12) is represented by a projection operator ${\hat{E}} = {\hat{P}}_{a_1, t_1} \otimes {\hat{P}}_{a_2, t_2} \otimes \ldots \otimes {\hat{P}}_{a_n, t_n}$ on $\mathcal{K}_{his}$. General history propositions are represented by projectors on $\mathcal{K}_{his}$ that are not factorized. Then, the set $\mathcal{V}$ of history propositions coincides with the lattice $\Lambda (\mathcal{K}_{his})$ of projection operators on $\mathcal{K}_{his}$. Hence, the logical operations on $\mathcal{V}$ coincide with the lattice operations of $\Lambda (\mathcal{K}_{his})$ [65].

The incorporation of dynamics into the histories description is rather intricate, and it requires a detailed analysis of the symmetries of the formalism. For continuous time, dynamics are implemented through an action operator that is defined on $\mathcal{K}_{his}$ [31].

The rule (13) does not define a probability measure on the set of history propositions. Probabilities can only be defined with respect to a given context. In the Copenhagen interpretation, the context is given by the measurement set-up, i.e., the choice of the different PVMs at different moments of time.

In decoherent histories, the context is expressed in terms of the abstract concept of a consistent set. To this end, we define the decoherence functional for a pair of histories $\alpha$ and $\beta$ of the form (13), as

$$\begin{aligned} d(\alpha , \beta ) = Tr\left( {\hat{C}}_{\alpha } {\hat{\rho }}_0 {\hat{C}}_{\beta }^{\dagger }\right) . \end{aligned}$$

(16)

The decoherence functional is extended to general history propositions by the requirements of

• linearity: $d(\alpha \; \text{ OR }\; \gamma , \beta ) = d(\alpha , \beta ) + d(\gamma , \beta )$, for $\alpha \; \text{ AND }\; \gamma = \emptyset$, and

• hermiticity: $d(\alpha , \beta ) = d(\beta , \alpha )^*$.

Then, d defines a bilinear map on $\mathcal{V} \times \mathcal{V}$.

Consider a set $\mathcal{W}$ of history propositions that are mutually exclusive and exhaustive. We call $\mathcal{W}$ a consistent set, if it satisfies the consistency condition

$$\begin{aligned} \text{ Re } \; d(\alpha , \beta ) = 0, \; \alpha \ne \beta , \end{aligned}$$

(17)

for all $\alpha , \beta \in \mathcal{W}$. Then, the diagonal elements of the decoherence functional define a probability measure on $\mathcal{S}$,

$$\begin{aligned} \text{Prob}_{\mathcal{W}}(\alpha) = d(\alpha, \alpha),\,\text{for}\,\text{all}\; \; \alpha \in {\mathcal{W}}. \end{aligned}$$

(18)

Hence, in quantum theory we do not have a complete probability function on $\mathcal{V}$, but a family of partial probability functions ${\text{Prob}}_{\mathcal{W}}$, each corresponding to a different consistent set $\mathcal{W}$.

Energy Conservation in $\Psi \Phi$I Theories

In this section, we explore the issue of energy conservation in $\Psi \Phi$I theories. We consider classical processes, because energy conservation is precisely formulated in classical mechanics. However, some of the results can be appropriately generalised also for quantum systems.

1.1 Energy Conserving Dynamics

We first consider deterministic processes. They describe dynamical systems, i.e., systems with time evolution determined by a set of differential equations. These equations can be interpreted as a flow on a differentiable manifold that plays the role of the state space. For MBI models, the state space is of the form $\Gamma _{\Phi }\times \Gamma _{\Psi }$, where $\Gamma _{\Phi }$ contains the physical degrees of freedom and $\Gamma _{\Psi }$ contains the mental degrees of freedom.

We describe the physical degrees of freedom in terms of classical mechanics, so that energy conservation is well-defined. Let us denote the points of $\Gamma _{\Phi }$ by $\xi ^a$, for some discrete index a. Time evolution is given by Hamilton’s equation

$$\begin{aligned} {\dot{\xi }}^a = \mathcal{P}^{ab} \partial _bH, \end{aligned}$$

(19)

where H is the Hamiltonian, i.e., a function on $\Gamma _{\Phi }$ whose values correspond to the usual notion of energy.

The Poisson tensor $\mathcal{P}^{ab}$ in Eq. (19) is antisymmetric, non-degenerate and satisfies Jacobi’s identity

$$\begin{aligned} \mathcal{P}^{ad}\partial _d\mathcal{P}^{bc} + \mathcal{P}^{cd}\partial _d\mathcal{P}^{ab} + \mathcal{P}^{bd}\partial _d\mathcal{P}^{ca} = 0. \end{aligned}$$

(20)

The antisymmetry of $\mathcal{P}$ guarantees the conservation of energy, since

$$\begin{aligned} {\dot{H}} = \partial _aH {\dot{\xi }}^a = \mathcal{P}^{ab} \partial _a H \partial _bH = 0. \end{aligned}$$

(21)

The most general dynamical system on $\Gamma _{\Phi }\times \Gamma _{\Psi }$, compatible with Eq. (19), is of the form

$$\begin{aligned} {\dot{\xi }}^a= & {} \mathcal{P}^{ab}(\xi ) \partial _bH(\xi ) + G^a(\xi , y) \end{aligned}$$

(22)

$$\begin{aligned} {\dot{y}}^i= & {} F^i(y) + J^i(\xi , y), \end{aligned}$$

(23)

where we denoted the points of $\Gamma _{\Psi }$ by $y^i$; i is a discrete index that labels the mental degrees of freedom. The vector $F^i$ on $\Gamma _{\Psi }$ describes self-dynamics on $\Gamma _{\Psi }$. Interaction is described by the vector fields $G^a$ on $\Gamma _{\Phi }$ and $J^i$ on $\Gamma _{\Psi }$.

Eq. (22) implies that

$$\begin{aligned} {\dot{H}} = G^a \partial _a H. \end{aligned}$$

(24)

Hence, energy is conserved if $G^a \partial _a H = 0$, i.e., if $G^a$ is tangent to the surfaces of constant energy. To motivate our subsequent analysis, let us first assume that physical degrees of freedom satisfy some form of Hamilton equations, even in the presence of MBI. This implies that $G^a$ is a Hamiltonian vector field for each y, i.e., that

$$\begin{aligned} G^a(\xi , y) = \mathcal{P}^{ab} \partial _b S(\xi , y) \end{aligned}$$

(25)

for some scalar function S. Hence, energy is conserved if $\mathcal{P}^{ab}\partial _aS \partial _bH = 0$.

A Hamiltonian system with a large number of degrees of freedom may have constants of the motion other than the Hamiltonian. However, it is highly implausible that we can ever relate such constants, when they are defined for very different systems, for example, a room with one person inside and a concert hall with one thousand persons.

We expect that energy-conserving dynamics of sufficient generality are possible only if S depends on $\xi$ solely through the Hamiltonian, i.,e., if $S(\xi , y) = \phi (H(\xi ), y)$ for some function $\phi : {\pmb R} \times \Gamma _{\Psi }\rightarrow {\pmb R}$. Then, the evolution equation on $\Gamma _{\Phi }$ becomes

$$\begin{aligned} {\dot{\xi }}^a = \tilde{\mathcal{P}}^{ab}( \xi , y) \partial _bH(\xi ) \end{aligned}$$

(26)

where $\tilde{\mathcal{P}}^{ab}(\xi , y) = [1 + \phi '[H(\xi ), y]]\mathcal{P}^{ab}(\xi )$, and the prime denotes the derivative of $\phi$ with respect to its first argument. Hence, energy is conserved if the mental degrees of freedom are coupled to the physical degrees of freedom through an antisymmetric tensor $\tilde{\mathcal{P}}^{ab}$.

The tensor $\tilde{\mathcal{P}}^{ab}$ is not an actual Poisson tensor, because it does not satisfy the Jacobi identity, Eq. (20). However, a small modification of our previous analysis can justify a Poisson tensor $\tilde{\mathcal{P}}^{ab}$ in Eq. (26), leading to a $\Psi \Phi$I theory with a much more interesting mathematical structure.

To this end, let us assume that there is a submanifold $\Omega \subset \Gamma _{\Psi }$ that corresponds to the mental vacuum of Sect. 5.2, i.e., if $y \in \Omega$ then no mental processes are present. Consider now a Poisson tensor $\tilde{\mathcal{P}}^{ab}$ that satisfies $\tilde{\mathcal{P}}^{ab} = \mathcal{P}^{ab}$, for any $y \in \Omega$. Then, Eqs. (26) and (23) define a dynamical system for MBI that reduces to the classical equations of motions in absence of mental processes. Hence, they define a deterministic $\Psi \Phi$I theory with energy conservation.

The difference from our previous analysis is that $G^a$ is not a Hamiltonian vector field, i.e., Eq. (25) does not hold. Instead, $G^a = \Delta \mathcal{P}^{ab} \partial _b H$, where $\Delta \mathcal{P}^{ab} = \tilde{\mathcal{P}}^{ab} - \mathcal{P}^{ab}$.

The implementation of energy conservation constraints only one of the two equations that describe the dynamical system, namely, Eq. (26) Thus, it persists even if the term $F^i(y)$ of Eq. (23) is a ‘random force’, i.e., if the values of $F^i$ at each moment of time are distributed probabilistically. Thus, energy conservation also applies to stochastic $\Psi \Phi$I theories—in fact it is completely insensitive to the dynamics of the mental degrees of freedom—, as long as Eq. (26) is satisfied.

The same structure can be employed in order to define energy-conserving dynamics for MBI theories, where matter is treated quantum mechanically. Schrödinger’s equation on a Hilbert space $\mathcal{H}$ is equivalent to Hamilton’s equation on the projective Hilbert space $\mathcal{PH}$ [66]. Hence, the above analysis can straightforwardly be transferred into a quantum context.

1.2 Generalised Energy

In mechanical systems, energy conservation arises as a consequence of the time-translation symmetry, i.e., the fact that dynamics are unchanged under a transformation that moves the time of the various events. It is plausible that the time translation symmetry of the extended dynamics (22–23) corresponds to a conserved quantity $K(\xi , y)$ that reduces to the Hamiltonian H in a limit where the mind–body interaction term can be ignored. For example, $K(\xi , y)$ may be of the form $L(y) + H(\xi )$, where L(y) is invariant under the self dynamics of the mental degrees of freedom, $F^i \partial L = 0$. Then,

$$\begin{aligned} {\dot{K}} = G^a \partial _a H + J^i\partial _i L, \end{aligned}$$

(27)

and the conservation of K leads to a relation between the coupling terms $G^a$ and $J^i$: $G^a \partial _a H + J^i\partial _i L = 0$. If this condition holds, the Hamiltonian is part of a more general conserved quantity K. Since K is assumed to follow from the time-translation symmetry of the dynamics, its values should be identified with energy. Thus, L is a new form of energy associated to mental processes, and the notion of energy has to be extended in order to account for mind–body interactions.