# The Potential of a Thick Present through Undefined Causality and Non-Locality

## Abstract

**:**

## 1. Introduction

**T**is considered, from which spacetime emerges in a sequence of space-like foliations.

_{k}_{k}are interpreted as logically consistent “memory-loops”, in which the information potential of an “open choice” (temporal order or spatial position) persists along the succession of the present instants.

## 2. Existence in a Thick Present

#### 2.1. Understandings on the Nature of Time

#### 2.2. Identifying a Quantum of Evolution

**T**related to a thick space-like foliation, bounded by −T and +T and derived from a time-symmetric superposition of perspectives (from a near past and a near future) on the emerging spacetime.

_{k}**T**from which spacetime emerges as a space-like foliation at 2kT.

_{k}**T**represents the potential of the kth space of events and possibilities in a space-like foliation of our universe bounded within (2k − 1)T and (2k + 1)T. In a QIS picture, its duration 2T is intended as a spacetime information “sampling rate”. The idea of a maximum rate of change connected to the inverse of the Planck time has been elaborated in Ref. [28], where it has been proven to be compatible with Relativity.

_{k}#### 2.3. Conclusion on Presentism and Open Challenges

**T**plays a crucial role in the emergence of spacetime, and a better description of how this potential is encoded in the thickness of the present is needed. Moreover, recent experiments have shown that undefined causality (entanglement in time order) is possible.

_{k}## 3. Undefined Orders in Imaginary Times

#### 3.1. Causality and Logical Consistency

#### 3.2. Imaginary Closed Paths

_{F,O}) needed at the speed of light from any quanta of space, the imaginary paths traced along (ict

_{F,O}) (or ict, for short) define and trace an imaginary Minkowski space within the thick present.

## 4. CTC, Entanglement, and Non-Locality

#### 4.1. The Potential Hidden in a Choice

_{k}as a potential of superposed values of the outcome of the choice and eventually a state with no identification of any event. In this sense, a CTC developing in the imaginary time represents an undefined causality in the thick present and an entanglement in the time order of the potential events along the closed path.

_{k}as spacetime “memory-loops”, able to encode the information potential of a logically consistent choice in the current space of events and possibilities, of which the outcome is open at the most fundamental level.

#### 4.2. Chasing Non-Local Information

**T**, the identification of a choice represents an information potential encoded and persisting in the superposition of the outcomes.

_{k}**T**, an entanglement in the time order (as UCO) as well as among particles in different spatial locations (EPR pairs) can be equivalent to a CTC, described as a logically consistent “memory-loop” able to encode the information potential of an open choice (offered in the entanglement) that precludes under or over-determined solutions. Loops are, actually, the most basic circuits for information storage, and CTC@T

_{k}_{k}can be considered as a spacetime “virtual memory” to encode the potential in

**T**.

_{k}**S**to

**R**is entanglement-breaking “along the path from

**S**to

**R**”. When the channels are in series but with an undefined order thanks to a controller qubit, we are implicitly assuming that the coherence with the controller has persisted along the time of traversal of both channels (as well as in the connections to go back to

**S**before entering the next channel), defining coherent imaginary paths, persisting at

**R**before any measurement on the controller. The resulting CTC in the imaginary time (coherent along the thick present instants even after the transmission over both channels) defines the UCO through the controller qubit and represents the additional quantum resource where the information is encoded identically along the communication (as for the channel ${C}_{+}$ in [47]). When the channels are in parallel, after the time of traversal there are no CTCs at

**S**or

**R**that could be used as an available quantum resource to encode the transmitted information, given that the entanglement on the “which way” would only define a causal branch, eventually selecting a single channel which would still be entanglement-breaking.

## 5. Towards a Holographic Perspective

**T**along an imaginary time of motion. This potential represents the information of entanglement, manifested as undefined causality and non-locality and encoded in the thick present as CTC or “memory loops”.

_{k}**T**as equivalent to a pair of symmetric bulk regions, in a time-symmetric description from (2k − 1)T to 2kT and from (2k + 1)T to 2kT, respectively, of which the space foliation is the common boundary at 2kT.

_{k}**T**in terms of the entanglement among the quanta of space, towards a full comprehension of its encoding in the symmetric bulks and of its relation with the gravitational potential.

_{k}**T**), space (emerging in each present instant along an imaginary time), and entanglement (as causally undefined or spatially non-local information of correlation among imaginary quanta of space encoded in the consistent superposition of outcomes of an open choice), we could maybe dare to conjecture, as limited Flatlanders, the following synthesis:

_{k}## 6. Synthesis and Outlook

**T**encoding the kth space of events and possibilities through a time-symmetric description, from (2k − 1)T to (2k + 1)T. The potential

_{k}**T**has been pictured as a logically consistent information evolving along the present instants, coherent with what happened and what could happen.

_{k}**T**.

_{k}**T**has been proposed as an elementary quantum of action and the “fastest event” to evaluate differences, acting as a reference to consistently compare any relativistic perspective on spacetime among observers.

_{k}**T**has been conceptually extended to the information expressed in the wave function of an elementary particle.

_{k}_{k}. The additional needed research on this path has been left to a dedicated contribution.

**T**that encodes through CTCs the information of entanglement (as undefined causality and non-locality in the emerging space foliation) seems a promising starting framework for the interpretation of our universe in terms of information. The hope is that future works on this path may offer additional insights into the possible ontological nature of information in the emergence of spacetime, towards a proper quantum description of gravity and a more profound understanding of our universe.

_{k}## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

C-NOT | Controlled-NOT quantum gate. It operates on a quantum register consisting of 2 qubits C and S (Controller C and Target S) and flips the qubit S if and only if $|C\u27e9=|1\u27e9$. Used to create entangled pairs (EPR particles) |

CTC | Closed Time-like Curves. Closed paths in spacetime. No events occur on their path, and time has no causal or thermodynamic orientation along them. They represent an information potential in the superposition of the outcomes of an open choice (e.g., direction of travel along the curve). When developing between −T and +T in the thick present, they cross the imaginary time axis in distant quanta of space that result connected (non-locality, entanglement in space). When developing in the imaginary space foliation they represent indefinite orders of traversed quanta of space (undefined causality, entanglement in time). They are considered in the paper as “memory loops”. |

EPR | Einstein–Podolsky–Rosen. Reference to the 1935 famous contribution on the incompleteness of QM. Intended in the paper as the EPR paradox, a synonym of entanglement and entangled particles (EPR pairs) |

ER | Einstein–Rosen bridge. Quantum wormhole connecting far regions of spacetime. Intended in the paper as entanglement between distant quanta in the imaginary space foliation (non-local information potential) |

GR | General Relativity (theory of) |

ICT | Information and Communication Technologies |

ict | Imaginary coherent time. Described as the imaginary time of motion (it) at the speed of light (c) between distant quanta of space. It is proposed in the paper as an imaginary axis of spatial distance and, in the absence of additional information potential, it defines a flat space foliation emerging in the present |

OPT | Operational Probabilistic Theories. Description of QM from first principles based on Information Theory |

PI | Path Integral formulation of QM |

QG | Quantum Gravity |

QIS | Quantum Information Science |

QM | Quantum Mechanics |

T_{k} | Quantum potential existing in the thick present (at the current evolution cycle, after 2kT instants from the origin of the universe). Time symmetric description from (2k – 1)T and (2k + 1)T of the entanglement in space (non-local potential in the imaginary space foliation emerging at 2kT along ict) and of the entanglement in time (undefined causality and indefinite time orders) |

TSVF | Two State Vector Formalism. Time-symmetric description of QM |

UCO | Undefined Causal Orders. Entanglement in time (order) |

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**Figure 1.**Identification of the thick present as the current thick space-like foliation and corresponding kth elaboration cycle of the quantum information

**T**, from which spacetime is considered to be emerging.

_{k}**Figure 2.**Controlled quantum SWITCH reproducing an UCO described as a XOR function, implemented as a C-NOT quantum gate. The entanglement of the Controller qubit C with the Target S at the point ⊕ allows the superposition of paths in which “A(B) is met first and B(A) is not”, and consequently the UCO. The entanglement in the C-NOT gate can be seen as the information potential of a choice instantiated in the instant of the interaction and of which the answer is undetermined.

**Figure 3.**Superposition of the imaginary paths of the particle $\left(iv{\tau}_{|1\u27e9}\oplus iv{\tau}_{|0\u27e9}\right)$ in a space-like foliation at a given instant after the time of traversal Δτ. The result can be described as the superposition of a forward- and backward-evolving wave in the imaginary time of motion $\left(iv{\tau}_{|C\u27e9}\right)$ and a CTC closed between C and any two points A and B on the circuit.

**Figure 4.**The imaginary space emerges from the information potential in the thick present, and it is described through an imaginary time. CTCs in the thickness are an expression of non-local potential.

**Figure 5.**Successive snapshots of the imaginary space foliation, from the EPR pair generation to the spin measurement at Alice’s location. The information potential persists along the successive instants through the entanglement/CTC as long as it is undetermined. When A defines a contextual outcome in her measurement (blue arrow), the state of the particle directed towards B is coherently defined so that the information keeps a global logical consistency within the thick present.

**Figure 6.**Graphics illustrating the space foliation at 2kT in the thick present as the boundary between two symmetric bulk regions extending from (2k − 1)T to 2kT and from (2k + 1)T to 2kT. The bulks encode the information

**T**(intended in a holographic perspective as an entanglement among quanta of space) from which spacetime emerges. CTCs in the thickness of the present represent the non-local potential; CTC developing in the imaginary time are equivalent to undefined orders of traversed points.

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Capurso, A.
The Potential of a Thick Present through Undefined Causality and Non-Locality. *Entropy* **2022**, *24*, 410.
https://doi.org/10.3390/e24030410

**AMA Style**

Capurso A.
The Potential of a Thick Present through Undefined Causality and Non-Locality. *Entropy*. 2022; 24(3):410.
https://doi.org/10.3390/e24030410

**Chicago/Turabian Style**

Capurso, Alessandro.
2022. "The Potential of a Thick Present through Undefined Causality and Non-Locality" *Entropy* 24, no. 3: 410.
https://doi.org/10.3390/e24030410