The generalized Hamilton principle and non-Hermitian quantum theory

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems. On this basis, we have given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, we have given the generalized Hamiltonian function for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics.


I. INTRODUCTION
In quantum mechanics, each classical physical quantity corresponds to an operator, and the operator has a real eigenvalue, which is guaranteed by the Hermitian operator. The Hermitian operator has always been generally considered to represent observable measurements. In fact, in quantum mechanics, it is only necessary to guarantee the observability of the mechanical quantity, but not to guarantee that its operator must be Hermitian, that is, observable measurement may also be non-Hermitian. In 1947, in order to solve the divergence problem in the field theory, Pauli used the indeterminate metric to put forward the theory of the non-Hermitian operator and its self-consistent inner product, which was derived from a field quantization method proposed by Dirac [1,2]. In order to maintain the unitary nature of the S matrix, Lee and Wick applied the non-Hermitian view to quantum electrodynamics [3]. Later, in different fields, numerous studies have proved that under certain conditions, the non-Hermitian Hamiltonian quantum has a real number energy spectrum [4][5][6][7]. In 1998, the author Bender proposed the space-time inverse symmetry (PT symmetry) quantum mechanics, which made the non-Hermitian quantum mechanics have a great leap forward [8,9]. The non-Hermitian P T symmetric Hamilton do not violate the physical principles of quantum mechanics and have real eigenvalues. Over the past decade P T symmetric quantum theory has been developed into a variety of studies, including field theory and high-energy particle physics. Recently, preliminary studies on P T symmetric systems under optical structures have been carried out.
The quantum theory of non-Hermitian is described dissipative systems and open systems, their unique properties have attracted fast growing interest in the last two decades [10][11][12][13][14], especially those empowered by parity-time symmetry [15]. While the non-Hermitian quantum theories is still under intense investigation, its application in different fields has led to a plethora of findings, ranging from nonlinear dynamics [16], atomic physics [17], photonics [18], acoustics [19], microwave [20], electronics [21], to quantum information science [22].
In this paper, we have given the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems. On this basis, we have given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, we have given the generalized Hamiltonian function for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics.

II. THE HAMILTON PRINCIPLE FOR THE CONSERVATIVE SYSTEM
In a mechanical system, the constraints that limit its position and speed can be written as equations the number of constraints equations are h. For the mechanical system of N free particles, their degree of freedom is 3N , when they are restricted by h constraints of equation (1), we can select 3N − h generalized coordinates q 1 , q 2 , · · · , q 3N −h , the position vector r i can be written as the generalized coordinates q i constitute the configuration space of 3N − h dimension the virtual displacement are the generalized velocity is˙ with Eq. (6), we can calculate the virtual work of active force F i , it is the generalized force Q j is If the generalized force Q j is conservative force, Eq. (7) becomes where U is the potential energy. In rectangular coordinates, there is and the component is In the following, we should study the system motion from time t 1 to t 2 , the T is the system kinetic energy, there is where with v i = d ri dt , we have Thus, Eq. (13) becomes i.e., According and If the variation of two endpoints are zero, there are and Eq. (18) becomes As the kinetic energy T is determined by the speed of each moment, there is When the active force F is conservative force, the work it does can be expressed as potential energy U , it is Then, Eq. (21) becomes i.e., or Where the Lagrange function L = T − V , and the action S = t2 t1 Ldt. The Eq. (25) or (26) is the Hamilton principle for the conservative system.

III. THE GENERALIZED HAMILTON PRINCIPLE FOR THE NONCONSERVATIVE SYSTEM
When the active forces include both conservative force F 1 and nonconservative force F 2 , we have Substituting Eqs. (27) and (28) into (21), there are We define generalized Lagrange function L, and Eq. (31) becomes The Eq. (33) is called the generalized Hamilton principle for the nonconservative force system, it is different from the Hamilton principle (25) for the conservative force system, the Eq. (33) contains the work of nonconservative force, and the variation is inside the integral sign. From Eq. (7), we can give the work of nonconservative forces F 2i (i = 1, 2, · · · , N ), it is When there is a single nonconservative force F 2 , there is So, when there are both conservative force F 1 and nonconservative force F 2 for the system, the generalized Lagrange function is the generalized action is and the generalized Hamilton principle is When there is only nonconservative force F 2 , and there is not conservative force F 1 for the system, the generalized Hamilton principle is and the generalized Lagrange function is

IV. THE GENERALIZED HAMILTON PRINCIPLE FOR THE HEAT EXCHANGE SYSTEM
In the mechanical, the change rate of energy is For a microcosmic particle, when it exchanges heat Q with the outside world, there is and the radiant force should be produced, it is When the microcosmic particle absorb heat, dQ dt > 0, the radiant force is F = −k v. When the microcosmic particle deliver heat, dQ dt < 0, the radiant force is F = k v. The Eq. (43) should be changed to the following formula i.e., then the radiant force is a nonconservative force. When a microcosmic particle exchanges heat with the outside world, its generalized Lagrange function is the generalized Hamiltonian function for the heat exchange system is and the generalized Hamilton principle for the heat exchange system is

V. THE GENERALIZED LAGRANGE EQUATION AND GENERALIZED HAMILTON FUNCTION FOR THE NONCONSERVATIVE SYSTEM
(1) The generalized Lagrange equation for the nonconservative system For the nonconservative system, the generalized Lagrange function is i.e., the variation of L is where δω 2 = F 2 · δ r, δ r = ∂ r ∂qi δq i and F 2 · ∂ r ∂qi = F 2i .
Substituting Eq. (52) into the generalized Hamilton principle (38), there is Obviously, there is Substituting Eq. (54) into (53), we have d dt The Eq. (55) is the generalized Lagrange equation for the nonconservative system.
(2) The generalized Hamilton function for the nonconservative system When L and w 2 do not include time, the time derivative of L is where w 2 = F 2 · d r and F 2 = ∂w2 ∂ r .
Using Eq. (56), we have or As then The H is called the integral of generalized energy, or generalized Hamilton function for the nonconservative force system.
(3) The invariance of L and the conserved quantity With Eqs. (52) and (55), we have By the invariance of L (δL = 0), we can obtain the conserved quantity for the nonconservative system which is the same as the conservative system.

VI. THE GENERALIZED LAGRANGE EQUATION AND GENERALIZED HAMILTON FUNCTION FOR THE HEAT EXCHANGE SYSTEM
(1) The generalized Lagrange equation for the heat exchange system In Eq. (47), the generalized Lagrange function for the heat exchange system is In section 8 (Eq. (91)), we have given the microcosmic heat Q = T S, then the Eq. (63) becomes i.e., When L and T do not include time, the variation of L is When the δq i is arbitrary, we obtain The Eq. (68) is the generalized Lagrange equation for the heat exchange system.
(2) The generalized Hamilton function for the heat exchange system When L and T do not include time, the time derivative of L is With Eq. (69), we have or ∂L ∂q iq i − L + T S = H = constant. For then The H is called the integral of generalized energy, or generalized Hamilton function for the heat exchange system.

(3) The invariance of L and the conserved quantity
In Eqs. (66) and (68), we have By the invariance of L (δL = 0), we can obtain the conserved quantity for the heat exchange system which is the same as the conservative system.
In the above, we have given the generalized Hamilton principle for the nonconservative force and the heat exchange system. On this basis, we further given the generalized Lagrange function and generalized Hamilton function for the nonconservative force and the heat exchange system. With the results, we shall study the non-Hermitian quantum theory for the nonconservative force and the heat exchange microcosmic system.

VII. THE NON-HERMITIAN QUANTUM THEORY FOR THE NONCONSERVATIVE FORCE SYSTEM
With the generalized Hamilton principle and generalized Lagrange function, we will deduce the non-Hermitian quantum theory for the nonconservative force system by the approach of path integral, and the path integral formula is In Eq. (76), the generalized Lagrange function L is where the force F is the nonconservative force. The Eq. (76) gives the wave function at a time t ′ in terms of the wave function at a time t. In order to obtain the differential equation, we apply this relationship in the special case that the time t ′ differs only by an infinitesimal interval ε from t. For a short interval ε the action is approximately ε times the Lagrangian for this interval, we have where A is a normalization constant. Substituting Eq. (77) into (78), there is In macroscopic field, the frictional force and adhere force are non-conservative force, and the non-conservative force F is directly proportional to velocity v, their directions are opposite, i.e. F = −k v. In microcosmic field, atomic and molecular can also suffer the action of non-conservative force. In the experiment of Bose-Einstein condensates, the atomic Rb 87 , N a 23 and Li 7 can be cooled in laser field, since they get the non-conservative force from the photons. Substituting F = −k v into Eq. (79), we get The quantity ( r− − → r ′ ε ) 2 appears in the exponent of the first factor. It is clear that if − → r ′ is appreciably different from r, this quantity is very large and the exponential consequently oscillates very rapidly as − → r ′ varies. When this factor oscillates rapidly, the integral over − → r ′ gives a very small value. Only if − → r ′ is near r do we get important contributions. For this reason, we make the substitution − → r ′ = r + η with the expectation that appreciable contribution to the integral will occur only for small η, and we obtain Now we have so that Substituting Eq. (83) into (81), we have After more complex calculation, it is obtained that where the Hamiltonian H isĤ Obviously, the Hamiltonian H is non Hermitian. The Detailed derivation can see the Ref. [23].

VIII. THE NON-HERMITIAN QUANTUM THEORY FOR THE THERMODYNAMICS
In classical mechanics, the energy of a macroscopic object is For a microcosmic particle, when it exchanges heat Q with the outside world. Using Eq. (48) or (73), the particle total energy should be the sum of kinetic energy, potential energy and thermal energy, namely, In thermodynamics, for the infinitely small processes, the entropy is defined as For the finite processes, it is At temperature T , when a particle has the microcosmic entropy S, it should has the thermal potential energy Q, and the Eq. (88) becomes Eq. (92) is the classical total energy of a microcosmic particle. In quantum theory, it should become operator form whereĤ = i ∂ ∂t ,p 2 = − 2 ∇ 2 andŜ is the microcosmic entropy operator.
At the i − th microcosmic state, the classical microcosmic entropy S F i and S Bi for Fermion and Bose systems are and where k B is the Boltzmann constant, n i is the average particle numbers of particle in the i − th state. For the Fermion (Bose), n i ≤ 1 (n i ≥ 1).
In quantum theory, the classical microcosmic entropy should become operator. The microcosmic entropy operator depends on temperature, but it has no the dimension of temperature, and it is non-Hermitian operator because it has to do with heat exchange. Therefore, the microcosmic entropy operator includes the temperature operator T ∂ ∂T . Moreover, it has to do with the state distribution. For the Fermion and Bose systems, the microcosmic entropy operator S F i andŜ Bi of a particle in the i − th state can be written aŝ andŜ In addition, we can prove the following operator relation and According to Eqs. (98)-(100), we find that the operator T ∂ ∂T is non-Hermitian. Thus, the microcosmic entropy operators (96) and (97) are also non-Hermitian, and the total Hamilton operator (93) is non-Hermitian and spacetime inversion (P T ) symmetryĤ This is because the particle (atom or molecule) exchanges energy with the external environment, which is an open system, its Hamiltonian operator should be non-Hermitian.

IX. THE SCHRODING EQUATION WITH TEMPERATURE
With the canonical quantization, E = i ∂ ∂t , p = −i ∇, substituting Eq. (96) into (93), we can obtain the Schroding equation with temperature By separating variables we obtain By separating variables Ψ n ( r, T ) = Ψ n ( r)φ(T ), Eq. (105) can be written as and where E n = E 1n + E 2n with E 1n being the eigenenergy obtained by the Schroding equation (106) and E 2n being the eigenenergy obtained by the temperature equation (107), the n expresses the n − th energy level, n i is the average particle numbers of the i − th state in the n − th energy level, and f n is the degeneracy of the n − th energy level. From Eq. (107), we can obtain that its solution is and where the temperature wave function φ(T ) is with A being the normalization constant and T 0 being the temperature constant. The general solution of Eq. (102) is For a free particle, its momentum is p in the environment of temperature T . Because this is in the determinate state, i.e., the average particle numbers n i = δ ij , the free particle plane wave solution and total energy are ψ( r, t, T ) = Ae i ( p· r−Et+i and By the accurate measurement the hydrogen atom spectrum, we can determine the temperature constant T 0 . The hydrogen atom has only one electron outside the nucleus, the degeneracy of the n − th energy level is f n = n 2 . When where h is the Planck constant. By measurement transition frequency ν exp mn , we can determine the temperature constant T 0 . When the electron jumps from the first excited state (m = 2) to ground state (n = 1), the T o is .
The theory should be tested by the experiments.

X. CONCLUSIONS
The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems. On this basis, we have given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, we have given the generalized Hamiltonian function for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics.