# A five-dimensional perspective on the Klein–Gordon equation

## Abstract

We discuss the Klein–Gordon (KG) equation using a path-integral approach in 5D space–time. We explicitly show that the KG equation in flat space–time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schrödinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space–time picture. Finally, we apply our results to the problem of hydrogenic spectra and calculate the canonical sum of the hydrogenic atom.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22451165

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics

- Additional Journal Information:
- Journal Volume: 356; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; KLEIN-GORDON EQUATION; MANY-DIMENSIONAL CALCULATIONS; PATH INTEGRALS; PROBABILISTIC ESTIMATION; SCHROEDINGER EQUATION; SPACE-TIME

### Citation Formats

```
Breban, Romulus, E-mail: breban@gmail.com.
```*A five-dimensional perspective on the Klein–Gordon equation*. United States: N. p., 2015.
Web. doi:10.1016/J.AOP.2015.02.028.

```
Breban, Romulus, E-mail: breban@gmail.com.
```*A five-dimensional perspective on the Klein–Gordon equation*. United States. doi:10.1016/J.AOP.2015.02.028.

```
Breban, Romulus, E-mail: breban@gmail.com. Fri .
"A five-dimensional perspective on the Klein–Gordon equation". United States. doi:10.1016/J.AOP.2015.02.028.
```

```
@article{osti_22451165,
```

title = {A five-dimensional perspective on the Klein–Gordon equation},

author = {Breban, Romulus, E-mail: breban@gmail.com},

abstractNote = {We discuss the Klein–Gordon (KG) equation using a path-integral approach in 5D space–time. We explicitly show that the KG equation in flat space–time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schrödinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space–time picture. Finally, we apply our results to the problem of hydrogenic spectra and calculate the canonical sum of the hydrogenic atom.},

doi = {10.1016/J.AOP.2015.02.028},

journal = {Annals of Physics},

issn = {0003-4916},

number = ,

volume = 356,

place = {United States},

year = {2015},

month = {5}

}

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