Klein-Gordon Equation within the Real Hilbert Space Formalism

For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities $J_a^μ$ , with $a\in(-1,1)$ , and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density ${\cal J}_a^μ$ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both $J_a^μ$ and ${\cal J}_a^μ$ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike $J_a^μ$ the current density ${\cal J}_a^μ$ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of $J_a^μ$ . The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter $a$ is rational. It is the multiplicative group of positive real numbers if $a$ is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities $J_a^μ$, with $a\in(-1,1)$, and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density ${\cal J}_a^μ$ for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both $J_a^μ$ and ${\cal J}_a^μ$ tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike $J_a^μ$ the current density ${\cal J}_a^μ$ is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of $J_a^μ$. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter $a$ is rational. It is the multiplicative group of positive real numbers if $a$ is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.

arXiv:quant-ph/0312078Dec 9, 20034 citations

The Klein–Gordon Equation and Scalar Quantum Fields

Peter Woit

Springer eBooksJan 1, 20170 citations

Another form of the Klein-Gordon equation

Morton D. Kostin

Physics Letters ANov 1, 19877 citations

Quantum Mechanics of Klein-Gordon Fields I: Hilbert Space, Localized States, and Chiral Symmetry

Alí Mostafazadeh, Farhad Zamani

We derive an explicit manifestly covariant expression for the most general positivedefinite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. This expression involves a one-parameter family of conserved current densities J µ a, with a ∈ (−1,1), that are analogous to the chiral current density for spin half fields. The conservation of J µ a is related to a global gauge symmetry of the Klein-Gordon fields whose gauge group is U(1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein-Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.We derive an explicit manifestly covariant expression for the most general positivedefinite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. This expression involves a one-parameter family of conserved current densities J µ a, with a ∈ (−1,1), that are analogous to the chiral current density for spin half fields. The conservation of J µ a is related to a global gauge symmetry of the Klein-Gordon fields whose gauge group is U(1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein-Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.

Jan 1, 200631 citations

From the main equation to the Klein-Gordon equation

Austin Blaquière

Journal of Optimization Theory and ApplicationsJan 1, 197910 citations

For free Klein-Gordon fields, we construct a one-parameter family of conserved current densities

J_a^μ

, with

a\in(-1,1)

, and use the latter to yield a manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein-Gordon equation. Employing a recently developed method of constructing the Hilbert space and observables for Klein-Gordon fields, we then obtain the probability current density

{\cal J}_a^μ

for the localization of a Klein-Gordon field in space. We show that in the nonrelativistic limit both

J_a^μ

and

{\cal J}_a^μ

tend to the probability current density for the localization of a nonrelativistic free particle in space, but that unlike

J_a^μ

the current density

{\cal J}_a^μ

is neither covariant nor conserved. Because the total probability may be obtained by integrating either of these two current densities over the whole space, the conservation of the total probability may be viewed as a consequence of the local conservation of

J_a^μ

. The latter is a manifestation of a previously unnoticed global gauge symmetry of the Klein-Gordon fields. The corresponding gauge group is U(1) if the parameter

a

is rational. It is the multiplicative group of positive real numbers if

a

is irrational. We also discuss an extension of our results to Klein-Gordon fields minimally coupled to an electromagnetic field.