# Time and Causality: A Thermocontextual Perspective

## Abstract

**:**

## 1. Introduction

#### 1.1. The Problems of Time

#### 1.2. The Problem of Nonlocality

“…absolute determinism in the universe [and] the complete absence of free will. Suppose the world is super-deterministic, … the difficulty disappears. There is no need for a faster than light signal to tell particle A what measurement has been carried out on particle B, because the universe, including particle A, already “knows” what that measurement and its outcome will be”.[10]

#### 1.3. We Need a Better Conceptual Model

- The possibility of superposed live–dead cats (Copenhagen interpretation [12]);
- Exponentially splitting worlds (many-worlds interpretation [13]);
- Superdeterminism (classical statistical mechanics, relativity, and quantum hidden-variable theories [9]);
- Nonlocality (the nonlocal de Broglie–Bohm pilot wave theory [9]).

## 2. The Thermocontextual Interpretation of State

**TCI**) is an alternative to the existing HCF and subjective interpretations of physical reality. As with any conceptual model of physics, the TCI is an axiomatic system based on (1) empirically validated physical facts, (2) fundamental premises, and (3) a definition of perfect measurement.

- Empirical conservation laws (e.g., energy, momentum, charges, and quantum spin);
- Empirical laws of motion (e.g., Newton’s laws and quantum mechanics);
- Empirical laws of interaction (e.g., law of gravitation and Planck’s law of radiation).

#### 2.1. Postulates and Definitions of State

**Definition**

**1.**

_{a}, equals the positive temperature of the system’s actual surroundings with which it interacts or potentially interacts.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

_{gs}, is the ambient ground state’s potential work capacity as measured at the limit of absolute zero.

**Definition**

**6.**

_{sys}= E − Q

_{gs}.

**Definition**

**7.**

_{sys}− X.

**Definition**

**8.**

**Definition**

**9.**

_{a}.

_{a}, ΔX < 0.

_{gs}+ E

_{sys}= Q

_{gs}+ X + Q.

_{sys}), exergy (X), and ambient heat (Q) are all thermocontextual properties of state, measured and defined with respect to the system’s ambient ground state (definition four), which defines their zero values. The ambient ground state energy (Q

_{gs}) is defined by measurements in the limit of absolute zero.

_{a}are empirically given by:

#### 2.2. Entropy and Refinement

_{a}. As with ambient heat, the TCI entropy is a thermocontextual property of state that is defined relative to the system’s equilibrium ground state at the ambient temperature. The TCI resolves total entropy (diagonal vector in Figure 2) into two components: the ambient entropy (S

_{amb}, horizontal vector) and the entropy of refinement (S

_{ref}, vertical vector).

_{ai}to a final T

_{af}. The entropy of thermal refinement simply reflects a change in the ambient entropy due to a shift in the temperature scale for measuring entropy. If the ambient temperature is constant (T

_{ai}= T

_{af}) then S

_{ref}(vertical vector) equals zero and the ambient entropy and TCI entropy are equal.

_{af}, and dq is the incremental change in heat thermalized at temperature T. The first equality is based on definition nine and the second equality follows from Equation (2), with T

_{ai}= T and T

_{af}= T

_{a}. In both cases the changes are in response to the change in the ambient temperature prior to any heat transfer or other adjustments to the changed surroundings.

_{ai}= T

_{sys}), as T

_{af}and the temperature of re-thermalization approach absolute zero, the ambient heat and the ambient entropy also approach zero. In the limit of T

_{a}= 0 kelvins the ambient entropy (horizontal vector) is zero. The TCI entropy therefore equals the entropy of refinement, and we get:

_{ref}and S

_{TCI}are both equal to thermodynamics’ third law entropy, defined by the integral term. Conversely, the thermocontextual TCI entropy is a generalization of thermodynamics’ third law entropy, defined for the idealized special case of an ambient temperature of absolute zero.

#### 2.3. Classical and Quantum States

_{a}= T. The gas’s total energy is defined by the equilibrium temperature and pressure, and its thermocontextual properties all equal zero. The equilibrium gas defines the gas’s “e-contextual” state, where the TCI defines an e-contextual state by equilibrium with its ground state.

## 3. Thermocontextual Time

#### 3.1. Thermodynamic Time

_{TD}, given by:

_{q}) by:

_{0}is the system’s initial exergy, X is the exergy at reference time t

_{ref}as measured by a reference clock in the ambient surroundings, and t

_{unit}is a chosen unit of reference time. The exergy and thermodynamic time for a first-order kinetic system, for example, are given by:

_{ref}) = e

^{−}

^{λ}

^{tref}X

_{0}and t

_{q}= (λt

_{unit})t

_{ref},

_{0}, and as time advances toward infinity the system’s exergy approaches zero. As with exergy, thermodynamic time is a thermocontextual property of state. As a thermocontextual property, thermodynamic time is incompatible with and ignored by noncontextual interpretations of physics.

#### 3.2. Mechanical Time

_{m}), which we take as a coordinate of imaginary mechanical time. It is important to note that imaginary mechanical time does not involve a transformation of coordinates or a Wick rotation, it is simply a change in terminology and leaves all equations unchanged. For example, the TCI expresses the time-dependent quantum wavefunction for an isolated (fixed energy) and non-reactive quantum system by:

#### 3.3. System Time and Reference Time

_{qi}. A change in thermodynamic time (horizontal axis) describes the irreversible dissipation of exergy and transition into a more stable state of lower exergy.

_{r}in Figure 4B) is the time of relativity as measured by an external clock. It is the time by which we measure the advance of a light cone; it defines the arrow of relativistic causality. Effects always follow causes in reference time, even when they are connected over system time by time-symmetrical determinism. Reference time provides the time scale across which a system’s events are observed and velocities are measured. As with an observer’s clock, it ticks forward whether the system’s time proceeds reversibly or irreversibly.

## 4. Instantiation and Actualization

**Definition**

**8.1.**

**Definition**

**8.2.**

#### 4.1. Statistical Entropy

_{i}to each classical mechanical microstate, “i”. Each microstate is precisely defined by perfect measurement at absolute zero. Each p

_{i}expresses the subjective probability that a system exists in microstate “i”, and Gibbs entropy expresses an observer’s subjective uncertainty of a system’s actual microstate. Gibbs entropy is an informational entropy. Except for a constant multiple, it is identical to Shannon’s information entropy.

_{B}, Equation (10) also defines von Neumann entropy [20]. The summation for the von Neumann entropy of a mixed quantum state is over the individual component states’ wavefunctions, and the p

_{i}are their respective weightings. It follows that a “pure” quantum state, having a single wavefunction, has zero von Neumann entropy. As with Gibbs entropy, von Neumann entropy expresses an increase in an observer’s subjective uncertainty of a system’s actual state following wavefunction collapse of a pure state into a mixed state. Gibbs and von Neumann entropies are information entropies; they are not physical properties of state.

_{i}in (11) are the objective probabilities that microstate potentiality “i” will be randomly selected when the system’s entropy is reset to zero, at which point the system reverts to a single measurable potentiality. A classical system’s entropy is reset to zero when the system’s entropy is entirely transferred to the new surroundings. A quantum system’s entropy is reset to zero when its superposed state vector is projected onto one of its new basis vectors. This describes the random collapse of an indefinite positive-entropy superposed state to a definite zero-entropy state. The probabilities in (11) are independent of observation or an observer’s knowledge, and the TCI entropy is objectively defined.

#### 4.2. Instantiation and Wavefunction Collapse

_{B}T

_{a}), where k

_{B}is the Boltzmann constant and T

_{a}is the temperature of thermalization. Equating the two probabilities implies a connection between ambient temperature and time. The TCI, however, does not recognize a connection between ambient temperature and mechanical time because microstate potentialities are intrinsically random, which means that they cannot be indexed as a function of time.

#### 4.3. Actualization and Measurement

## 5. Entanglement and Nonlocality

#### 5.1. Entanglement—A Mechanical Illustration

#### 5.2. Quantum Nonlocality—Just the Facts

#### 5.3. Nonlocality and Bell Locality—An Explanation

_{q},it

_{m1}) (Figure 7) the interaction transfers entropy to the surroundings and derandomizes the photon pair. This resets the entangled pair’s entropy to zero and instantiates one of their potentialities as a definite zero-entropy microstate (see Figure 5). Entanglement by parallel polarization constrains the pair’s instantiated and entangled microstate to either $\left(\updownarrow \updownarrow \right)$ or $\left(\leftrightarrow \leftrightarrow \right)$. All of this deterministically occurs over an interval of time-symmetrical mechanical time, it

_{m0}–it

_{m1}, within an instant of thermodynamic time at t

_{q}(Figure 7).

_{q}to t

_{q’}and sets mechanical time to a new interval of time symmetry, it

_{m’0}–it

_{m’1}.

_{q’},it

_{m’0}) and Bob transmits his result to Alice via a light signal. Alice receives the results recorded by Bob at point A’ and time (t

_{q’},it

_{m’1}) and is able to verify that Bob’s results are correlated with hers. Alice’s and Bob’s observations of their measurement results, Bob’s transmission of his results, and Alice’s recording of his results are conducted across time-symmetrical mechanical time within the instant of thermodynamic time t

_{q’}.

_{rA}, and her recording of Bob’s measurement result at t

_{rA’}are reversibly linked within an instant of thermodynamic time via A↔A’↔B (Figure 7), but her reference time ticks irreversibly forward. Even though Bob and Alice cannot receive the other’s results until after they conduct their own measurements, they each conclude the other’s measurement was conducted simultaneously with their own and that the measurement results were instantaneously correlated.

## 6. Summary and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Perfect measurement is a reversible transformation from a system’s initial state to its ambient ground state reference. Perfect reversible measurement involves transfers of exergy (X) and ambient heat (Q) to the surroundings. Reversing the process restores the system’s initial pre-measurement state.

**Figure 3.**Mechanical and reference times. (

**A**) Plot shows the trajectory of a one-dimensional harmonic oscillator over parallel advances in mechanical and reference times. (

**B**) Mechanical and reference times decouple when the oscillator encounters an elastic barrier momentarily inserted at time t

_{b}. The oscillator and mechanical time, as an index of the oscillator’s position, reverse direction and retrace the trajectory over mechanical time. The trajectory in B shows the oscillator’s position across reference time, which continues to advance without bound.

**Figure 4.**Complex system time and reference time. (

**A**) Shows the complex system time plane, spanned by real-valued thermodynamic time (horizontal axis) and imaginary mechanical time (vertical axis). (

**B**) Shows the irreversible advance in reference time over reversible intervals of system time at a fixed thermodynamic time, t

_{qi}, and between irreversible transitions over intervals of thermodynamic time, Δt

_{qi}.

**Figure 5.**Instantiation of metastable microstate. The microstate initially has positive entropy and comprises multiple measurable potentialities (dots). Instantiation involves the transfer of ambient heat and entropy to the surroundings. This reduces the microstate’s entropy to zero and randomly instantiates a single zero-entropy potentiality (black dot).

**Figure 6.**Perfect measurement. Perfect measurement is a reversible open system transition from an instantiated zero-entropy microstate to its ambient ground state reference. During perfect measurement the initial state’s exergy actualizes a record of the transition on the surroundings.

**Figure 7.**Instantaneous correlation of spatially separated measurements. The Figure spans space (projected onto the horizontal axis) and mechanical time (left axis). Mechanical time spans two reversible intervals at two distinct instants of thermodynamic time, separated by the irreversible actualization of photon measurement results. Superimposed on the diagram are light cones advancing across the reference time for the measurement events at A and B (right axis). Each light cone shows the domain of causality from A or B within the constraints of relativity.

Energy Component | n-Particle Ideal Gas | Hydrogen Atom at Temperature T | Thermocontextual? |
---|---|---|---|

State Description | Equilibrium state (T,P; V = nk _{B}T/P) | $\mathsf{\Psi}\left(\mathrm{T}\right)={\displaystyle {\displaystyle \sum}_{\mathrm{i}}}{\mathrm{c}}_{\mathrm{i}}\left(\mathrm{T}\right){\mathsf{\psi}}_{\mathrm{i}}\mathrm{where}{\displaystyle {\displaystyle \sum}_{\mathrm{i}}}{\left|{\mathrm{c}}_{\mathrm{i}}\left(\mathrm{T}\right)\right|}^{2}=1$ | No |

Energy (total) | E = nk_{B}T | $\langle \mathrm{E}\left(\mathrm{T}\right)\rangle ={\displaystyle {\displaystyle \sum}_{\mathrm{i}}}{\mathrm{E}}_{\mathrm{i}}\times {\left|{\mathrm{c}}_{\mathrm{i}}\left(\mathrm{T}\right)\right|}^{2}={\displaystyle {\displaystyle \sum}_{\mathrm{j}}}{\mathrm{p}}_{\mathrm{j}}\left(\mathrm{T}\right){\mathrm{E}}_{\mathrm{j}}{}^{\left(1\right)}$ | No |

Q_{gs} (ground state energy) | Q_{gs} = nk_{B}T_{a} | Q_{gs} = $\langle \mathrm{E}\left({\mathrm{T}}_{\mathrm{a}}\right)\rangle ={\displaystyle {{\displaystyle \sum}}_{\mathrm{j}}}{\mathrm{p}}_{\mathrm{j}}\left({\mathrm{T}}_{a}\right){\mathrm{E}}_{\mathrm{j}}$ | Yes |

E_{sys} (system energy) | E_{sys} = E − Q_{gs} | $\langle {\mathrm{E}}_{\mathrm{sys}}\left(\mathrm{T}\right)\rangle =\langle \mathrm{E}\left(\mathrm{T}\right)\rangle -{\mathrm{Q}}_{\mathrm{gs}}={\displaystyle {\displaystyle \sum}_{\mathrm{j}}}{\mathrm{E}}_{\mathrm{j}}\left({\mathrm{p}}_{\mathrm{j}}\left(\mathrm{T}\right)-{\mathrm{p}}_{\mathrm{j}}\left({\mathrm{T}}_{\mathrm{a}}\right)\right)$ | Yes |

Q = T_{a}S_{TCI} = T _{a}S_{ref}^{(2)} (ambient heat) | $\mathrm{Q}={\mathrm{T}}_{\mathrm{a}}\left({{\displaystyle \int}}_{{\mathrm{T}}_{\mathrm{a}}}^{{\mathrm{T}}_{\mathrm{sys}}}{\mathrm{C}}_{\mathrm{v}}\left(\mathrm{T}\right)\frac{\mathrm{d}\mathrm{T}}{\mathrm{T}}\right)$ | $\langle \mathrm{Q}\left(\mathrm{T}\right)\rangle ={\mathrm{T}}_{\mathrm{a}}\left({{\displaystyle \int}}_{{\mathrm{T}}_{\mathrm{a}}}^{\mathrm{T}}\left(\frac{\partial \mathrm{E}\left(\mathrm{T}\right)}{\partial \mathrm{T}}\right)\frac{\mathrm{d}\mathrm{T}}{\mathrm{T}}\right)$ | Yes |

X (exergy) | X = E_{sys}−Q | $\langle \mathrm{X}\left(\mathrm{T}\right)\rangle =\langle {\mathrm{E}}_{\mathrm{sys}}\left(\mathrm{T}\right)\rangle -\langle \mathrm{Q}\left(\mathrm{T}\right)\rangle $ | Yes |

S_{TCI} = Q/T_{a} (entropy) | ${\mathrm{S}}_{\mathrm{TCM}}=\frac{\mathrm{Q}}{{\mathrm{T}}_{\mathrm{a}}}={{\displaystyle \int}}_{{\mathrm{T}}_{\mathrm{a}}}^{{\mathrm{T}}_{\mathrm{sys}}}{\mathrm{C}}_{\mathrm{v}}\frac{\mathrm{d}\mathrm{T}}{\mathrm{T}}$ | $\langle {S}_{\mathrm{TCM}}\left(\mathrm{T}\right)\rangle =\frac{\mathrm{Q}\left(\mathrm{T}\right)}{{\mathrm{T}}_{\mathrm{a}}}={{\displaystyle \int}}_{{\mathrm{T}}_{\mathrm{a}}}^{\mathrm{T}}\left(\frac{\partial {\mathrm{E}}_{\mathrm{sys}}\left(\mathrm{T}\right)}{\partial \mathrm{T}}\right)\frac{\mathrm{d}\mathrm{T}}{\mathrm{T}}$ | Yes |

_{B}= Boltzmann constant. C

_{v}= volumetric heat capacity. Angle brackets indicate the expectation (and time-averaged) values. (1) The expectation energy value is defined by a weighted sum over the eigenfunction energies, E

_{i}, which typically includes degenerate and unresolvable eigenfunctions sharing the same energy. The TCI expresses the energy expectation value as a weighted sum over its discrete measurable microstate energies, E

_{j}, and its probabilities, p

_{j}. (2) The gas and hydrogen atom are related to their ambient states by a change in ambient temperature only, prior to any other changes. S

_{TCI}therefore equals S

_{ref}(vertical vector in Figure 2).

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Crecraft, H. Time and Causality: A Thermocontextual Perspective. *Entropy* **2021**, *23*, 1705.
https://doi.org/10.3390/e23121705

**AMA Style**

Crecraft H. Time and Causality: A Thermocontextual Perspective. *Entropy*. 2021; 23(12):1705.
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**Chicago/Turabian Style**

Crecraft, Harrison. 2021. "Time and Causality: A Thermocontextual Perspective" *Entropy* 23, no. 12: 1705.
https://doi.org/10.3390/e23121705