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On the basis of the canonical quantization procedure of a system defined on a cubic lattice, we propose a new method, in which resolutions of unity expressed in terms of eigenvectors are naturally provided, to find eigenvectors of field operators. By making use of fundamental ingredients thus obtained, we derive time-sliced path integral formulae for a massive vector field accompanied by a scalar field. Due to the indefinite metric of the Hilbert space upon which we define field operators, the action that appears in the path integral looks quite different from the classical one. Nevertheless, we will find that the effective action defined by introducing external sources results in the original action. By taking the effective action as the basis of consideration, we study the proper meaning of the covariance of a path integral.

The path integral method [

On the manifestly covariant path integral of the vector field, on which the Faddeev–Popov procedure [

Another issue we should like to consider is the meaning of the prescription

We consider here the quantization of a massive vector field accompanied by a Nakanishi–Lautrup

The paper is organized as follows: in

We begin with the Lagrangian proposed by Nakanishi [

We impose the periodic boundary condition (PBC) on this lattice so that the integration by parts, for instance

If we introduce a new vector field

The equations of motion read

If we introduce

In the same way,

To achieve the canonical quantization, we choose

Canonical commutation relations are given by

The classical solution in Eq.

In the same way, we may put

We solve the expansions above to find:

The Schrödinger operators are obtained by setting

Note that we also have

The vacuum is defined to be destroyed by all

To formulate path integrals for a system, the fundamental ingredient we need is the resolution of unity. When we utilize the holomorphic representation, the resolution of unity will be expressed in terms of the coherent state. We try here to construct eigenvectors of field operators and shall find suitable expressions for the resolution of unity on the Hilbert space equipped with the indefinite metric, as we have seen in the previous subsection.

To obtain the field diagonal representation, let us introduce, for spatial components of the vector field,

Their commutation relation reads

By regarding these operators as creation and annihilation operators, we make use of the technique shown in the appendix to find eigenvectors of

We define the coherent state

If we write

We can check that the coherent state forms a complete set in the Fock space defined above for spatial components of the vector field. Furthermore, since the operators

We consider the operator

By integrating the

If we notice that

If we carry out the Gaussian integrals with respect to the

The inner product of these eigenvectors is given by

Decomposition of

We then define the coherent state

By writing

Eigenvectors of

The resolution of unity on the Fock space of the spatial component can be expressed as

By making use of the commutation relation

Here we should add a comment that the factor

Then, by making use of the relation

Leaving the consideration of the spatial components of the vector field, we now proceed to the construction of eigenvectors of

Since their commutation relation is given by

We first define the coherent state

The conjugate of this coherent state is given by

On this state,

The inner product between these coherent states is given by

We now define

By remembering the inner product

Note that the eigenvalue of

Again through Gaussian integrals over

To obtain eigenvectors of

Through a similar process as that for the eigenvectors of

The bra and ket vectors above satisfy

We thus again find that the eigenvalue of

The resolution of unity for this degree of freedom can be expressed in terms of these eigenvectors as

By making use of the commutation relation

We have thus obtained the desired eigenvectors of field operators to reflect the indefinite metric of the Hilbert space. Combining all the above results, we can now express the resolution of unity on this Hilbert space. To simplify the notation, we will write

In the same way, Eqs.

The inner products are then rewritten as

We have thus completed the preparation of our tools for formulating the path integral representation of the system described by the Hamiltonian in Eq.

In this section we consider the Euclidean path integrals of the system described by the Hamiltonian in Eq.

To formulate the path integral representation of the system, we rewrite the Hamiltonian in terms of Schrödinger operators as

We assume hereafter the rule of sum over repeated indices, including the index for lattice points. Since our concern is the quantum theory of the original vector field

We can therefore express the source terms above as

Note that we have defined the source terms in terms of Schrödinger operators.

By taking the effect of the source terms into account, we first evaluate a short-time kernel,

Upon finding that

Introducing

We may define Green’ s functions by

Covariance in the above Green’ s functions will be translated into that for Green’ s functions in the Minkowski space by putting

Covariance in Minkowski space will be seen more clearly by considering the effective action of the system under consideration. To this end, we go back to the generating functional

We then define the effective action through the Legendre transform by

Equations of motion for the classical solutions read

We are now able to write the effective action in terms of the Euclidean vector field

In order to see that the effective action obtained above reproduces the covariant action in the Minkowski space, let us now take the continuum limit by putting

Upon the transition to Minkowski space, according to the change

This is nothing but the continuum version of the action that corresponds to the original Lagrangian of Eq.

It must be noted here that the covariance both in the generating functional

So far we have formulated the path integral by utilizing eigenvectors of

Here, a comment is needed: the action above should be compared with the one given by Eq. (17) of Ref. [

Integrations with respect to the

Thus the covariance of the Euclidean action is evident, and we observe that the Euclidean path integral in terms of

As we will confirm at the end of this section, however,

It is straightforward to find that

Hence, by carrying out Gaussian integrals with respect to the

The calculation of the effective action from

Before closing this section, we have to add comments on eigenvalues of the

Our final comment on the path integral considered here is that our prescription is available only under the trace formula because we have introduced spatial components of

Through the previous arguments, our definition of the path integral deeply depends on the Euclidean technique. Within the framework of a Euclidean path integral, as far as the mass of the particle is kept to be

By putting

Here,

Note that we have introduced

We now introduce

We then obtain

By completing the square of

It is evident that the propagator for the vector field

Classical solutions are determined by

Taking these into account, we can easily rewrite the generating functional

By remembering the explicit form of

It is straightforward to find that

It will be interesting to compare the process above with the covariant formulation of the path integral for the system under consideration. On the basis of canonical commutation relations

By carrying out integrations with respect to

If we carry out the Fresnel integrals with respect to the

By defining classical solutions to satisfy

The manipulation above must, however, be corrected in the following points to be acceptable. First of all, the Fresnel integrals with respect to the

In the first half of this paper we have introduced a unified method of finding eigenvectors of field operators to obtain eigenvectors of a massive vector field accompanied with a scalar field, as well as those for their canonical conjugates, based on the Lagrangian proposed by Nakanishi [

Resolutions of unity thus obtained can be utilized to formulate path integrals for the vector field, keeping a good connection with the covariant canonical formalism. On the basis of these fundamental ingredients, we have considered path integrals of the vector field in Euclidean space and in Minkowski space in the second half of this paper. Because operators quantized with an indefinite metric require imaginary eigenvalues, the Euclidean path integrals of the system considered in this paper become well defined, but the manifest covariance of the action in the exponent of the path integral cannot be expected for the same reason in the Minkowski case. We needed to introduce

We may generalize the method of constructing the field diagonal representation shown in this paper to other cases such as the quantization of Lee–Wick models with indefinite-metric representation. The Euclidicity postulate may work as the guiding principle to formulate Euclidean path integrals, even for such systems. This will be discussed elsewhere.

We have had no chance to discuss the BRS invariance of the system because our system explicitly breaks the BRS symmetry due to the mass term of the vector field. It is, however, possible to extend the system to be part of the Higgs model [

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Let us begin with the Lagrangian

The corresponding Hamiltonian

To find the eigenvector of the position operator

By expressing

Then, by carrying out the integration with respect to

If we remember the relations

Integration of the right-hand side of Eq. (

Therefore, we can write

It is straightforward to check that these are eigenvectors of

In the same way, we can find that eigenvectors of the momentum operator

The inner products of the eigenvectors obtained above are given by

The completeness of these eigenvectors is written as

These are the fundamental ingredients for constructing path integrals by means of the time-slicing method.

Quantization of a system described by the Lagrangian

Note that the Heisenberg operator

Due to the commutation relation above, the norm of these eigenvectors is not always positive, and is given by

To remove the

The coherent state defined by

Similar to the positive norm case, we consider here

Integration of the right-hand side above clearly exhibits that

With this observation, we find that the eigenvector of

The eigenvalue of

In the same way, we find that the eigenvector of

The eigenvalue of

Inner products of the eigenvectors thus obtained are given by

The completeness of these eigenvectors is written as

We make use of the expressions above to formulate a time-sliced path integral for the Hamiltonian given at the start of this appendix. To do so, we first consider the short-time kernel

We now set

Let us now consider the effect of an external source. To this end, we introduce

After dividing the time interval

We can compare this result with the corresponding one for a harmonic oscillator of the usual commutation relation

Note that the integral of the quadratic term of the external source has the opposite sign in Eq. (

We now set

By defining