Relativistic probability densities for location
Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ ( x , t ) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes $\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\r...
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| Format: | Article |
| Language: | English |
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IOP Publishing
2023-01-01
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| Series: | Journal of Physics Communications |
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| Online Access: | https://doi.org/10.1088/2399-6528/acddcc |
| _version_ | 1797798479301443584 |
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| author | Joshua G Fenwick Rainer Dick |
| author_facet | Joshua G Fenwick Rainer Dick |
| author_sort | Joshua G Fenwick |
| collection | DOAJ |
| description | Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ ( x , t ) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes $\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\rm{i}}{\omega }_{{\boldsymbol{k}}}t)$ yield normalized functions ψ ( x , t ) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ _μ ( x , t ) = ( ℏ /2i)[ ψ ^+ ( x , t )∂ _μ ψ ( x , t ) − ∂ _μ ψ ^+ ( x , t ) · ψ ( x , t )]. However, in the case of bosonic fields, the wave packets ψ ( x , t ) are nonlocally related to the corresponding relativistic quantum fields ϕ ( x , t ), and therefore the canonical local energy-momentum densities ${ \mathcal H }({\boldsymbol{x}},t)=c{{ \mathcal P }}^{0}({\boldsymbol{x}},t)$ and ${ \mathcal P }({\boldsymbol{x}},t)$ differ from ℘ _μ ( x , t ) and appear nonlocal in terms of the wave packets ψ ( x , t ). We examine the relation between the canonical energy density ${ \mathcal H }({\boldsymbol{x}},t)$ , the canonical charge density ϱ ( x , t ), the energy pseudo-density $\tilde{{ \mathcal H }}({\boldsymbol{x}},t)=c{\wp }^{0}({\boldsymbol{x}},t)$ , and the Born density ∣ ψ ( x , t )∣ ^2 for the massless free Klein–Gordon field. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case: in spite of their nonlocal mathematical relations, they are mutually local in the sense that their maxima do not deviate beyond a common position uncertainty Δ x . Indeed, they are practically indistinguishable in cases where we would expect a normalized quantum state to produce particle-like position signals, viz. if we are observing quanta with momenta p ≫ Δ p ≥ ℏ /2Δ x . We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density ${ \mathcal H }({\boldsymbol{x}},t)/E$ provides a suitable measure for positions of bosons, whereas normalized charge density ϱ ( x , t )/ q provides a suitable measure for fermions. |
| first_indexed | 2024-03-13T04:04:20Z |
| format | Article |
| id | doaj.art-81a9fdb65a314ed98cc1095f6c44f1cc |
| institution | Directory Open Access Journal |
| issn | 2399-6528 |
| language | English |
| last_indexed | 2024-03-13T04:04:20Z |
| publishDate | 2023-01-01 |
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| spelling | doaj.art-81a9fdb65a314ed98cc1095f6c44f1cc2023-06-21T09:16:33ZengIOP PublishingJournal of Physics Communications2399-65282023-01-017606500610.1088/2399-6528/acddccRelativistic probability densities for locationJoshua G Fenwick0https://orcid.org/0009-0008-7435-2216Rainer Dick1https://orcid.org/0000-0001-8740-6991Department of Physics and Engineering Physics, University of Saskatchewan , 116 Science Place, Saskatoon, SK S7N 5E2, CanadaDepartment of Physics and Engineering Physics, University of Saskatchewan , 116 Science Place, Saskatoon, SK S7N 5E2, CanadaImposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ ( x , t ) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes $\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\rm{i}}{\omega }_{{\boldsymbol{k}}}t)$ yield normalized functions ψ ( x , t ) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ _μ ( x , t ) = ( ℏ /2i)[ ψ ^+ ( x , t )∂ _μ ψ ( x , t ) − ∂ _μ ψ ^+ ( x , t ) · ψ ( x , t )]. However, in the case of bosonic fields, the wave packets ψ ( x , t ) are nonlocally related to the corresponding relativistic quantum fields ϕ ( x , t ), and therefore the canonical local energy-momentum densities ${ \mathcal H }({\boldsymbol{x}},t)=c{{ \mathcal P }}^{0}({\boldsymbol{x}},t)$ and ${ \mathcal P }({\boldsymbol{x}},t)$ differ from ℘ _μ ( x , t ) and appear nonlocal in terms of the wave packets ψ ( x , t ). We examine the relation between the canonical energy density ${ \mathcal H }({\boldsymbol{x}},t)$ , the canonical charge density ϱ ( x , t ), the energy pseudo-density $\tilde{{ \mathcal H }}({\boldsymbol{x}},t)=c{\wp }^{0}({\boldsymbol{x}},t)$ , and the Born density ∣ ψ ( x , t )∣ ^2 for the massless free Klein–Gordon field. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case: in spite of their nonlocal mathematical relations, they are mutually local in the sense that their maxima do not deviate beyond a common position uncertainty Δ x . Indeed, they are practically indistinguishable in cases where we would expect a normalized quantum state to produce particle-like position signals, viz. if we are observing quanta with momenta p ≫ Δ p ≥ ℏ /2Δ x . We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density ${ \mathcal H }({\boldsymbol{x}},t)/E$ provides a suitable measure for positions of bosons, whereas normalized charge density ϱ ( x , t )/ q provides a suitable measure for fermions.https://doi.org/10.1088/2399-6528/acddccrelativistic wave functionsBorn ruleprobability densities |
| spellingShingle | Joshua G Fenwick Rainer Dick Relativistic probability densities for location Journal of Physics Communications relativistic wave functions Born rule probability densities |
| title | Relativistic probability densities for location |
| title_full | Relativistic probability densities for location |
| title_fullStr | Relativistic probability densities for location |
| title_full_unstemmed | Relativistic probability densities for location |
| title_short | Relativistic probability densities for location |
| title_sort | relativistic probability densities for location |
| topic | relativistic wave functions Born rule probability densities |
| url | https://doi.org/10.1088/2399-6528/acddcc |
| work_keys_str_mv | AT joshuagfenwick relativisticprobabilitydensitiesforlocation AT rainerdick relativisticprobabilitydensitiesforlocation |