Aq CHRONOLOGICAL PRODUCTS , ENERGY-MOMENTUM TENSORS OF SCALAR FIELDS FOR PARTICLE GENERATIONS , AND INDEFINITE METRICS

The solutions of generalized Klein-Gordon equations are considered. The generalizations of the Klein-Gordon equation allow one to derive convergent integrals for the Green functions of these equations. The generalized equations are presented as products of the operators for the Klein-Gordon equation with different masses. The solutions of derived homogeneous equations (total fields) are sums of fields corresponding to particles with the same values of the spin, the electric charge, the parities, but with different masses. Such particles are grouped into the kinds (families, dynasties) with members which are the generations. The chronological products of the scalar fields for kinds of particles are obtained at arbitrary quantity of the generations. These chronological products are the causal Green functions of generalized Klein-Gordon equations. The Lagrangians for the generalized Klein-Gordon equations of arbitrary order are derived. These Lagrangians are used to obtain the energy-momentum tensors for the particle kinds at arbitrary quantity of generations. It is shown that the generalized Hamiltonians (for total fields) have got positive eigenvalues for all the generations. These results are derived with the use of the indefinite metrics.

spinless particle and corresponding projection operator.The particle propagator of the spinless particle in coordinate representation is the causal Green function of the Klein-Gordon equation (the Klein-Gordon-Fock equation).It means that the Green functions of the Klein-Gordon equation and other wave equations for any spin particles are infinite improper four-fold integrals with respect to momentum variables.In Refs.[9,10] it is shown that corresponding fourfold integrals for the Klein-Gordon and Dirac equations, as well as triple integral for the Yukawa potential, diverge.To eliminate these divergences the integrals for the Green functions have been generalized in Refs.[9,10].New Green functions correspond to equations with higher partial derivatives.Among these equations the generalized Klein-Gordon equation is very important, as many features of generalized equations may be seen.The ( )  x  solutions of the generalized Klein-Gordon equations can be presented as sum of solutions of the Klein-Gordon equations for different particle masses.These sums correspond to some sets of particles.Therefore, values related to the solutions of the generalized Klein-Gordon equations can have some feature in a comparison with similar values for the Klein-Gordon equations.In particular, it is of importance to study Lagrangians and energy-momentum tensors for generalized equations.
The aim of present paper is the study of some topics related to the solutions of the generalized Klein-Gordon equations.Similar topics for one field are well known [1][2][3][4][5][6][7][8].So, it is shown that the chronological product of quantized free solutions for the generalized Klein-Gordon equation is the causal Green function of this equation.In terms of the total fields the Lagrangians, the energy-momentum tensors, and a generalized Hamiltonian are derived.In particular, the question on the positive determination of the generalized Hamiltonian is investigated.In addition, conditions, which permit one to derive some general relations for the solution of the Klein-Gordon equations, are studied.

RELATION OF CHRONOLOGICAL PRODUCTS FOR FIELDS AND GREEN FUNCTIONS
It is known that the   -function and the   G x x  Green functions of differential equation can be expanded in the set of eigenfunctions for the operator of this differential equation [11].Accordingly to Refs.[2][3][4][5][6] the vacuum average of the chronological product for the scalar fields is related to the   The generalized Klein-Gordon equations and their   For the chronological product of the   A x operator and the   B y operator usual definition can be exploited: The discontinuous function   0 x  can be presented by the improper integral We propose that the total   x  quantized field is given by where k  are some phases.The k  phases can give parameters of mixing for composite states.The ( ) k x  field is the scalar quantized free field of the k number generation.
We use the indefinite metrics.The covariant normalization condition, the non-zero commutators of the ( ) k a p  ( ( ) k b p  ) annihilation operators for particles (antiparticles) and the ( ) ) creation operators are the generalizations of usual commutators for one particle: where 1 2 ,   are some discrete quantum numbers.The signs in the relations (5) agree with the signs of the k Using the formulae ( 2) and ( 4) the chronological product of the total scalar fields may be written as The vacuum matrix elements of the products for the annihilation operators and the creation operators equal the corresponding commutators (5).After the substitutions of these commutators in the chronological product (6) we can write: Now it can be used the representation (3) for discontinuous function.Then we obtain four-fold integral for (7) In Ref. [12] has been shown that for Note that each term in ( 9) is diverging triple integral.We change the variable of integration q in (8) by means of such variables: in the first term and 0 k q p    in the second term.Then the T-product (8) with the use of (9) can be written as The k  value (4) is positive and the 2 k i -value is the imaginary positive infinitesimal numbers.Therefore, we may denote the 2 k i value in (10) as the i value.Now the T-product of the total fields ( 10) is given by Thus, the chronological product of the total scalar fields ( 4) is related to the causal Green function (A.10) of the generalized Klein-Gordon equation (A.1), similarly to the (1) relation for one particle.This relation is the consequence of the indefinite metrics, expressed by the commutators (5), and the equality (9) at 3 N  .

Lagrangians for generalized Klein-Gordon equations
In Ref. [13] the Lagrangians for the generalized Dirac equations are derived.Therefore it is of interest to obtain the Lagrangian for the generalized Klein-Gordon equations.Operators of the generalized Klein-Gordon equations (A.1) are polynomials with respect to the  operator of d' Alembert (d'Alembertian).They can be written as The k S values are elementary symmetric functions [14].They equal: For these functions the formula can be written at k > 1   In the case of equal masses   As the operators of generalized Klein-Gordon equations (A.1) include the partial derivatives of the 2 N order and they are polynomials, Lagrangians for these equations must have polynomial structure.Let us denote In general, the Lagrangian can depend on the x coordinates, ( ) x  , ( ) x   fields, and their derivatives ( 16) for 1 n N   .Using the least action principle (the Ostrogradskii-Hamilton principle), the Ostrogradskii-Euler equations, which are generalizations of the Euler-Lagrange equations, can be derived (for example, see Refs.[15,16]).The equation (A.1) for the ( ) x  -field can be obtained by means of the variation of the Lagrangian L(x) with respect to the ( ) -field and their derivatives.The equation for the ( ) -field, similar to (A.1), can be derived by means of the variation of the L(x) Lagrangian with respect to the ( ) x  -field and their derivatives.In the terms of the definitions (16) the Ostrogradskii-Euler equation for the ( ) x  field may be written as The total Lagrangian equals The ( ) free L x -part of Lagrangian allows one to derive the homogeneous equation ( 1) from the equation ( 18) and the int ( ) L x -part leads to the right hand of the equation (A.1).The Lagrangian for homogeneous Klein-Gordon equation can be written as The Lagrangian of an interaction is given by A substitution of the ( 18)-( 20) Lagrangians into the Ostrogradskii-Euler equation (17) gives the generalized Klein-Gordon equation (A.1) at arbitrary quantity of generations in a kind (or a family or a dynasty).

Energy-momentum tensor for total fields
The Lagrangian (19) for free total fields (4) does not depend on the x coordinates explicitly.Therefore, we can expect that some conserved values (some first integrals of the equations) must exist.In the case of scalar field for one particle the conserved values constitute the energy-momentum tensor.In this case the Lagrangian includes the partial derivatives of the fields of first order.The Lagrangian (19) for the total fields (corresponding to the kind with N generations) includes the partial derivatives of the N order.The total derivatives of the Lagrangian (19) with respect to each coordinate equal After the substitution of the L   term from the equation (17) the derivatives (21) are given by The right hand of (22) must be presented as a sum of total derivatives with respect to the coordinates.In particular, the right hand of (22) equal: The use of total derivatives permits one to introduce the   E x  tensor, which is the generalization of the   T x  energy-momentum tensor for one particle (e.g., in Refs.[1][2][3][4][5][6][7][8]) .
The   E x  tensor is symmetric with respect to the  index and the  index.The four-dimension divergence of the   E x  tensor vanishes similarly to the   T x  energy-momentum tensor.Therefore, the vector must be conserved, i.e., components of this vector do not depend on the 0 x time.For quantized fields the components of this vector are operators.We shall consider matrix elements of these operators between one-particle states for the same particles.
At first, we calculate the contribution of the Lagrangian (19) to the  

d p d x i p p x a p a p b p b p i p p x S p p S d p A a p a p b p b p S m
The integration in (26) with respect to the spatial coordinates gives the     The derivatives of the Lagrangian ( 19) can be written as Now we find the vector of the four-momentum P  (25) e x p 2 exp exp 4 Again the integration with respect to spatial coordinates in (28) leads to the Then we can write using (A.5) After calculations of derivatives in (31) we must put 2 2 k q m  .Now the energy-momentum vector P for the kind of particles with the N generations is given by This vector can be presented in terms of normal products of the operators similarly to the Hamiltonian for one particle (Refs.[1][2][3][4][5][6][7][8]) The contribution to norm H for 1 k  in (33) equals the Hamiltonian of particle (Refs.[1][2][3][4][5][6][7][8]). The norm H value in (33) can be named as the generalized Hamiltonian or the total Hamiltonian or the Hamiltonian of particle generations.The total Hamiltonian (33) includes the sign factors 1 ( 1) k   in each term.Calculations of the eigenvalues for the total Hamiltonian with the use of the positive metrics give positive and negative values.Such calculations confirm known results that the particle energies are not positively determined for Lagrangians including higher derivatives of fields.Therefore, we consider the eigenvalues for the total Hamiltonian with the use of the indefinite metrics (5).Then for the particle with the q momentum , , of the k number generation the eigenvalues of the 4momentum may be derived . The Eqs. (34) are valid for particles.But similar equations can be written for antiparticles also.
Thus, the eigenvalues of the total Hamiltonian for all the particle generations are positive at the use of the indefinite metrics.This result is important.Therefore, it is of interest the consideration of different consequences of the indefinite metrics.

Consequences of indefinite metrics
The conserved P energy-momentum vector (32) has been derived from a condition that the Lagrangian (19) does not depend on x -coordinates explicitly.Now we calculate the vacuum average of the H -operator in (32) Thus, the vacuum average (32) of the energy is infinite, similarly to such value for one particle (Refs [1][2][3][4][5][6][7][8]). Note that vacuum average of the norm H total Hamiltonian (33) vanishes.As the (3) ( ) o   -value equals the spatial volume, the density for the vacuum average (32) of the energy is infinite too.
Further the norm P energy-momentum vector (33) will be considered without the norm -index (i.e., the operators in normal forms will be consider).
For arbitrary function of the P -momentum operators we may write In particular, the equalities are valid at an use of the indefinite metrics.Consider now the commutator of the total field and the operator of the total momentum (33) using the indefinite metrics: e a p e b p x P d q q a q a q b q b q x d p A e p e a p e b p i x The commutators (38) have the same forms as the commutators for one field and the Hamiltonian for one particle in Refs.[1][2][3][4][5][6][7][8].Therefore, an use of the commutator (38) allows one to derive Using ( 36) and (39), matrix elements may be written as The commutator of the particle annihilation operator and the total Hamiltonian (33) is given by Similarly, the commutators of the total Hamiltonian with the antiparticle annihilation operator and the creation operators can be derived: .
From the relations (41-43) we may conclude that indeed the ) operators are the annihilation operators of particle (antiparticle) with the 2 q  -momentum from the 2 k generation number and operators are similar creation operators of particle (antiparticle).Consider the operators and states with n particles and l antiparticles 1 1 2 2 1 1 , ; , ; ; , ; , ; ; , where   x  is the field and   x  is the current (the field source).In momentum space the differential operator in (A.1) is the polynomial of the N -degree.We consider the case of the polynomial with real non-negative different zeros at 2 where .In Ref. [9] it is shown that the case of equal masses in Eq. (A. e d q e d q G x q m q m q m P q where 2 ( ) N P q is the polynomial of the N degree with respect to 2 q .It is clear that the integrals in (A.4) can converge at 3 N  , i.e., when the order of the equation (A.1) is greater than or equals six.Consequently for each spinless particle two (or greater) particles with the same charges, isospin, C -and P parity, but different masses, must exist in addition.We may say that such particles are members of some set (a family or a kind or a dynasty).In Eq. (A.2) k is the number of the particle generation.We may assume that the quantity of members in kinds for the elementary particle is less than the quantity of member in kinds for the composite particle.Each particle belongs to some kind and some generation.
According to Refs.[9,10], the rational fraction in (A.4) can be written as The k A coefficients obey the relations: are named as total fields and fields of particle generations) are considered in Appendix.It is of interest to study the relation between the chronological product of the quantized   x  total fields, which are the solutions of the homogeneous equation (A.1), and the Green function of this equation.

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As the P  -vector does not depend on a time, the components of the 1 p momentum and the 2 p momentum must be equal assume that the quantity of generations is enough large for a convergence of the integrals in (26), similarly to the (9) relation.The factor in (26) vanishes for arbitrary k number generation.

2
equation (A.1) is the sum of the general solution of the corresponding homogeneous equation   free x  and partial solution   nh x  of non-homogeneous equation: k c and k c  are arbitrary constants.Thus,   free x  is the sum of the terms corresponding to particles with the same charges, parities, spin, but with different masses.Each term in (A.2) corresponding to number k is the solution of the homogeneous Klein-Gordon equation as 

7 )
Using the equality (A.5) we may write the Green functions (A.4) of Eq. (A.1) in the form 1) must be excluded.It was shown that the functions   free x  can include nonnormalizable terms if at least two masses are equal.Thus, the masses in the generalized Klein Gordon equation must be different.The N -number equals to the quantity of generations for spinless bosons and order of the equation (A.1) equals 2 N .The Green functions for the generalized Klein-Gordon equations (A.1) are given by