# Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription

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## Abstract

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## 1. Introduction: Current Status of Hamiltonian Theories

- Manifestly covariant, i.e., frame-symmetric. This means that the canonical variables, the Hamiltonian density and related functionals are necessarily set in 4-tensor form. As a consequence, the resulting Hamiltonian representation holds in arbitrary coordinate systems (i.e., GR-frames), which are mutually connected by local point transformations, i.e., diffeomorphisms of the type $r\to {r}^{\prime}={r}^{\prime}\left(r\right),$ with $r\equiv \left\{{r}^{\mu}\right\}$ and ${r}^{\prime}\equiv \left\{{r}^{\prime}{}^{\mu}\right\}$ denoting two arbitrary GR-frames.
- Variational, namely, as shown below (and as typical of classical Hamiltonian systems occurring in classical mechanics) the new Hamiltonian representation is prescribed via a suitable synchronous path-integral variational principle, expressed as a line-integral performed along a geodesic trajectory in terms of an invariant proper-time parameter, and for this reason referred to here as Hamilton variational principle.
- Unconstrained, i.e., the same Hamiltonian system can always be expressed in terms of an arbitrary independent set of canonical variables.

- Gauge-dependent, namely, such that both the Hamiltonian density and the corresponding Lagrangian density display definite gauge properties. However, the corresponding Euler–Lagrange equations (i.e., both the Hamilton and Lagrange equations) can be shown to be generally gauge-symmetric, namely, independent of the gauge itself. Nevertheless, by adopting a suitably-constrained variational principle the same Euler–Lagrange equations can also be equivalently modified in a way that they become gauge-dependent, namely to include an explicit additive gauge term which depends linearly on the CC. Accordingly, the same CC acquires the connotation of a 4-scalar gauge function which at the classical level remains undetermined.

#### 1.1. Background on the Cosmological Constant

#### 1.2. Statement of the Problem and Goals

## 2. The Cosmological Constant in Asynchronous Variational Principles

## 3. Manifestly-Covariant Hamiltonian Structure of SF-GR

## 4. New Hamiltonian Representation: Constraint-Free Hamilton Variational Principle

## 5. Constrained Hamilton Variational Principle

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Cremaschini, C.; Tessarotto, M.
Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription. *Symmetry* **2020**, *12*, 633.
https://doi.org/10.3390/sym12040633

**AMA Style**

Cremaschini C, Tessarotto M.
Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription. *Symmetry*. 2020; 12(4):633.
https://doi.org/10.3390/sym12040633

**Chicago/Turabian Style**

Cremaschini, Claudio, and Massimo Tessarotto.
2020. "Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription" *Symmetry* 12, no. 4: 633.
https://doi.org/10.3390/sym12040633