# ON THE DEFINITION OF CURRENTS AND THE ACTION PRINCIPLE IN FIELD THEORIES OF ONE SPATIAL DIMENSION

## Abstract

The properties of a model relativistic field theory in one space dimension are investigated. The model consists of a massless fermion field whose current is coupled to itself and also to a vector boson field with mass. By means of the action principle the effects of the coupling are simulated by functional derivatives with respect to external fields. These are carried out in closed form and yield an exact solution. It is found that the usual definition of the fermion current as the simple juxtaposition of the fermion field and its conjugate is ambigrous and does not lead to a relativistic two-vector. A more precise definition of the current, which remedies this situation, is obtained by taking the average of the limits of a nonlocal product of the fermion fields as the coordinates of those fields approach one another from a spacelike and the orthogonal timelike direction. Explicit examination of the Green's functions shows that those containing only one boson field or fermion current are consistent with the field equations, canonical commutation relations, and the definition of the current, but that this is not true for those containing two or more of these operators. This situation is clarified with themore »

- Authors:

- Publication Date:

- Research Org.:
- Harvard Univ., Cambridge, Mass. and Yale Univ., New Haven

- OSTI Identifier:
- 4110475

- NSA Number:
- NSA-18-010989

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York) (U.S.)

- Additional Journal Information:
- Journal Volume: Vol: 26; Other Information: Orig. Receipt Date: 31-DEC-64

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

- Subject:
- PHYSICS; ACTION PRINCIPLE; BEAMS; BOSONS; COMMUTATION RELATIONS; CONFIGURATION; CONSERVATION LAWS; DIFFERENTIAL EQUATIONS; ELECTRODYNAMICS; ELEMENTARY PARTICLES; ENERGY; EQUATIONS; FERMIONS; FIELD THEORY; GREEN FUNCTION; INTERACTIONS; INVARIANCE PRINCIPLE; LAGRANGIAN; MASS; MATHEMATICS; MATRICES; MECHANICS; MOMENTUM; MOTION; PARTICLE MODELS; QUANTUM MECHANICS; RELATIVITY THEORY; TENSORS; THERMODYNAMICS; VECTORS

### Citation Formats

```
Sommerfield, C M.
```*ON THE DEFINITION OF CURRENTS AND THE ACTION PRINCIPLE IN FIELD THEORIES OF ONE SPATIAL DIMENSION*. Country unknown/Code not available: N. p., 1964.
Web. doi:10.1016/0003-4916(64)90273-8.

```
Sommerfield, C M.
```*ON THE DEFINITION OF CURRENTS AND THE ACTION PRINCIPLE IN FIELD THEORIES OF ONE SPATIAL DIMENSION*. Country unknown/Code not available. doi:10.1016/0003-4916(64)90273-8.

```
Sommerfield, C M. Wed .
"ON THE DEFINITION OF CURRENTS AND THE ACTION PRINCIPLE IN FIELD THEORIES OF ONE SPATIAL DIMENSION". Country unknown/Code not available. doi:10.1016/0003-4916(64)90273-8.
```

```
@article{osti_4110475,
```

title = {ON THE DEFINITION OF CURRENTS AND THE ACTION PRINCIPLE IN FIELD THEORIES OF ONE SPATIAL DIMENSION},

author = {Sommerfield, C M},

abstractNote = {The properties of a model relativistic field theory in one space dimension are investigated. The model consists of a massless fermion field whose current is coupled to itself and also to a vector boson field with mass. By means of the action principle the effects of the coupling are simulated by functional derivatives with respect to external fields. These are carried out in closed form and yield an exact solution. It is found that the usual definition of the fermion current as the simple juxtaposition of the fermion field and its conjugate is ambigrous and does not lead to a relativistic two-vector. A more precise definition of the current, which remedies this situation, is obtained by taking the average of the limits of a nonlocal product of the fermion fields as the coordinates of those fields approach one another from a spacelike and the orthogonal timelike direction. Explicit examination of the Green's functions shows that those containing only one boson field or fermion current are consistent with the field equations, canonical commutation relations, and the definition of the current, but that this is not true for those containing two or more of these operators. This situation is clarified with the introduction of a noncovariant single-time current wherein the limit of the coordinates of the fermion fields is taken at a given time. A comparison of the matrix elements of this current with those of the covariant one shows that one of these must be considered as depending explicitly, in its definition, on the external fields. When the covariant current is so considered, the extra terms in its matrix elements coming from functional differentiation of the explicit field dependence are found to remove the inconsistency. The original use of the action principle in solving the model is re-examined and a corrected version of the Lagrange function that contains operators at one time only is derived. Because of the massless nature of the fermion field, there are expected to be both a current and a pseudocurrent that are conserved. It is found that neither of the currents (or corresponding pseudocurrents) is conserved; the actual conserved current and pseudocurrent are inferred from an examination of the matrix elements. It is then shown to what extent the action principle may be used to derive the conservation laws for them. It is also shown that three of the four components of the energy-momentum tensor may be derived directly from the Lagrange function without the knowledge of the effect of changing the direction of approach of the coordinates of the operators therein. These components are found to satisfy commutation relations that guarantee the relativistic invariance of the theory. The corrected Lagrange function was written down only after complete knowledge of the solution was at hand. The question of whether it is possible to know the Lagrange function without such knowledge is discussed. It is found that in the present case the answer is no, unless the current appearing in the field equations is defined in a manner different from the one used. A choice of the conserved current for this purpose is found to simplify some of the properties of the solution. This discussion is extended to more general models. (auth)},

doi = {10.1016/0003-4916(64)90273-8},

journal = {Annals of Physics (New York) (U.S.)},

number = ,

volume = Vol: 26,

place = {Country unknown/Code not available},

year = {1964},

month = {1}

}