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Birkhäuser Basel eBooks

The Schrödinger Equation can be a Boltzmann Equation

Masao Nagasawa

Published Jan 1, 1993• 0 citations
Publisher

Abstract

Let us consider the Schrödinger process {Xt,Q| determined by the triplet {P(s, x)c, μa, μb| (cf. Chapter 5), namely Q is Csiszar’s projection on the subset. Aa, b (see (8.4) below) of the renormalized process $$ \bar P $$ , or a diffusion process constructed by the method of Chapter 6 for a given function ϕ(t, x). In the latter case we take the creation and killing c = -Lϕ/ϕ induced by ϕ(t, x), and another function $$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } \left( {a,x} \right) $$ such that $$ \mu _a \left( x \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } \left( {a,x} \right)\varphi \left( {a,x} \right), $$ , and moreover assume the admissibility condition (5.12) so that Q = Qμa is the Csiszar projection of $$ \bar P $$ . This case occurs, if a solution ψ= eR+iS of the Schrödinger equation is given beforehand and if we define Õ = eR+s and $$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } = e^{R - S} $$ in terms of R and S of ψ= eR+iS.

DOI
10.1007/978-3-0348-8568-3_8

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Schrödinger equations and diffusion theory

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Preface.- I Introduction and Motivation.- II Diffusion Processes and their Transformations.- III Duality and Time Reversal of Diffusion Processes.- IV Equivalence of Diffusion and Schrodinger Equations.- V Variational Principle.- VI Diffusion Processes in q-Representation.- VII Segregation of a Population.- VIII The Schrodinger Equation can be a Boltzmann Equation.- IX Applications of the Statistical Model for Schrodinger Equations.- X Relative Entropy and Csiszar's Projection.- XI Large Deviations.- XII Non-Linearity Induced by the Branching Property.- Appendix.- References.- Index.

Birkhäuser-Verlag eBooks•Jan 1, 1993•0 citations

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Time-dependent quantum Green's functions

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Introduction Mathematically, given a linear differential operator L x , one encounters the solution to the inhomogeneous differential equation Here ρ ( x ) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let G ( x, y ) is the Green's function. This is a function of x with y a parameter. Take G ( x, y ) to satisfy the same boundary conditions as φ ( x ). Then since An example of L x is, of course, the Schrödinger operator We take ρ ( x ) = V ( x ) ψ ( x , t ), ψ ( x , t ) being the wave function and V ( x ) the potential operator. We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al. , 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000). One- and two-time quantum Green's functions and their properties Let us introduce the creation operator ψ † ( r, t ) and annihilation operator ψ ( r, t ) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles: The Hamiltonian operator for the particles is and the number density of particles at r t is the operator Note that r = ( r 1 … r N ) for (1, 2, 3, …, N ) and V (| r − r ′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.

Cambridge University Press eBooks•Apr 16, 2009•0 citations

Thermodynamic Selection, Projection-Constrained Schrödinger Dynamics, and Vacuum Admissibility in String/M-Theory (Document 182 // SPEC-386)

Torjusen Marius Egerhei

Document 182 (SPEC-386) in the Monograph Series on Causal Inertia, Epistemic Governance & Field-Theoretic Realization. A deep and pervasive fallacy persists in our understanding of complex systems: the illusion that discarding a theoretical possibility is a frictionless event. We have long assumed that rejected branches—whether unused configurations in quantum mechanics or unrealized universes in string theory—simply evaporate into a parallel void. Yet conceptual generation costs nothing; it is selection itself that demands payment. Document 182 establishes this as an absolute physical principle, wherein infinite potential is reduced from a mathematical luxury to a thermodynamic transaction. That which cannot be admitted by the system is not covertly stashed away; it is extinguished, and this reduction of freedom leaves an indelible, thermodynamic scar upon the physical world. This cost is rigorously quantified via Landauer's principle, which compels the system to settle the energetic account for every restriction of its degrees of freedom. When uninhibited, free dynamics is confronted with an uncompromising projection gate, non-viable dynamical freedom is brutally stripped away. This eliminated structure represents no passive absence, but an active selection residual that enforces unavoidable dissipation. The entropy balance permits no loopholes: the very work of constraining reality into a specific, permissible state consumes thermodynamic currency (ΔStotal = ΔSdyn + ΔSsel ≥ 0). Reality therefore emerges not as the aggregate of all that could exist in theory, but strictly as that which can afford to be chosen. At the quantum level, this mechanism is formalized through Projection-Constrained Schrödinger Dynamics (PCSE), where the evolution of the wave function is subjected to explicit structural constraint (iℏ ∂t ψ = Hψ − iℏλ(I − ΠK)ψ). Within the admissible subspace, unitarity and phase coherence are strictly preserved under Hamiltonian control. However, the instant intrinsic dynamics attempts to leak into forbidden configurations, it is countered by irreversible, dissipative damping. A fundamental, continuous tension arises between the coherent generation of possibilities and the system's unyielding demand for legality—a balance wherein any state violating the projection operator is exponentially driven to extinction. This uncompromising demand for viability fundamentally cleanses cosmology's most overcrowded landscape, specifically the unwieldy moduli space of string and M-theory vacua (> 10500). Rather than defaulting to the obscure premise that an infinity of parallel universes must co-exist to salvage mathematical consistency, the framework establishes an absolute boundary between theoretical solution space and physical realization. A physical universe cannot persist merely as an ephemeral equation; it must manifest as a fixed point capable of withstanding its own projection over time (x☆ ∈ Fix(ΠK ∘ Φ)). Only states anchored in the forward-invariant viability kernel are granted physical form. The remainder remains strictly solution space—barred from existence by a universe that never ceases to demand thermodynamic coverage for all that is called real. Core Theoretic & Quantitative Modules: Thermodynamic Selection & Minimal Landauer Bound: Free dynamics x't+1 = Φ(xt) generates candidate states, while realized dynamics xt+1 = ΠK(x't+1) is enforced by an idempotent projection operator. The selection residual ΔK(xt) = Φ(xt) − ΠK(Φ(xt)) represents discarded dynamical freedom. Whenever ΔK ≠ 0, N bits of physically encoded structure are erased, imposing the dissipative lower bound Qmin ≥ N kB T ln 2 and non-negative entropy production ΔStotal = ΔSdyn + ΔSsel ≥ 0. Hamiltonian Energy Formulation: Effective system energy HΠK = (χ/2) dF2(Kγ, Kref) + BA + (ζ/2) |OL(γ)|2 + Hirr attains its absolute minimum when kernel, barrier, witness, and irreversibility conditions are satisfied simultaneously: Esel = (χ/2) ||OK||2 + BA + Hirr. Projection-Constrained Schrödinger Dynamics (PCSE): iℏ ∂t ψ = Hψ − iℏλ(I − ΠK)ψ. Hamiltonian and unitary within admissible subspace K; illegal components exterior to K are damped non-unitarily at rate λ. For invariant subspaces ([ΠK, H] = 0), exponential convergence is exact: ||q(t)||2 = e−2λt ||q(0)||2. When [ΠK, H] ≠ 0, a dynamic equilibrium balances generation and dissipation. Vacuum Admissibility in String/M-Theory: On moduli space M, the cardinal distinction holds: Realization ≠ Solution; Realization ⊆ Solution Space. Physical vacua are selected as fixed points under forward-invariant viability: x☆ ∈ Fix(ΠK ∘ Φ) ∩ Γphys, where Γphys = { x0 ∈ M | Φt(x0) ∈ K ∀ t ≥ 0 }. Conventional unconstrained dynamics is recovered in the zero-projection limit (OK = 0). Epistemic Demarcation (AGENTS.md): Verified via L1 deterministic Python reference engine (thermodynamic_quantum_selection_engine.py) with 10/10 unit tests PASS (0.002s) under the cardinal doctrine INSPIRED_BY != IMPLEMENTS != PROVES.

•Oct 8, 2026•0 citations

Schrodinger's equation derived from the collision-less Boltzmann's equation

Tarek Halabi

Withdrawn: Needed more work.

arXiv:1007.1825•Jul 12, 2010•0 citations
Let us consider the Schrödinger process {Xt,Q| determined by the triplet {P(s, x)c, μa, μb| (cf. Chapter 5), namely Q is Csiszar’s projection on the subset. Aa, b (see (8.4) below) of the renormalized process Pˉ\bar PPˉ , or a diffusion process constructed by the method of Chapter 6 for a given function ϕ(t, x). In the latter case we take the creation and killing c = -Lϕ/ϕ induced by ϕ(t, x), and another function φ\lower0.5em⌢(a,x)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } \left( {a,x} \right)φ\lower0.5em⌢​(a,x) such that μa(x)=φ\lower0.5em⌢(a,x)φ(a,x),\mu _a \left( x \right) = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } \left( {a,x} \right)\varphi \left( {a,x} \right),μa​(x)=φ\lower0.5em⌢​(a,x)φ(a,x), , and moreover assume the admissibility condition (5.12) so that Q = Qμa is the Csiszar projection of Pˉ\bar PPˉ . This case occurs, if a solution ψ= eR+iS of the Schrödinger equation is given beforehand and if we define Õ = eR+s and φ\lower0.5em⌢=eR−S\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varphi } = e^{R - S}φ\lower0.5em⌢​=eR−S in terms of R and S of ψ= eR+iS.
Preface.- I Introduction and Motivation.- II Diffusion Processes and their Transformations.- III Duality and Time Reversal of Diffusion Processes.- IV Equivalence of Diffusion and Schrodinger Equations.- V Variational Principle.- VI Diffusion Processes in q-Representation.- VII Segregation of a Population.- VIII The Schrodinger Equation can be a Boltzmann Equation.- IX Applications of the Statistical Model for Schrodinger Equations.- X Relative Entropy and Csiszar's Projection.- XI Large Deviations.- XII Non-Linearity Induced by the Branching Property.- Appendix.- References.- Index.
Introduction Mathematically, given a linear differential operator L x , one encounters the solution to the inhomogeneous differential equation Here ρ ( x ) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let G ( x, y ) is the Green's function. This is a function of x with y a parameter. Take G ( x, y ) to satisfy the same boundary conditions as φ ( x ). Then since An example of L x is, of course, the Schrödinger operator We take ρ ( x ) = V ( x ) ψ ( x , t ), ψ ( x , t ) being the wave function and V ( x ) the potential operator. We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al. , 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000). One- and two-time quantum Green's functions and their properties Let us introduce the creation operator ψ † ( r, t ) and annihilation operator ψ ( r, t ) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles: The Hamiltonian operator for the particles is and the number density of particles at r t is the operator Note that r = ( r 1 … r N ) for (1, 2, 3, …, N ) and V (| r − r ′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.
Document 182 (SPEC-386) in the Monograph Series on Causal Inertia, Epistemic Governance & Field-Theoretic Realization. A deep and pervasive fallacy persists in our understanding of complex systems: the illusion that discarding a theoretical possibility is a frictionless event. We have long assumed that rejected branches—whether unused configurations in quantum mechanics or unrealized universes in string theory—simply evaporate into a parallel void. Yet conceptual generation costs nothing; it is selection itself that demands payment. Document 182 establishes this as an absolute physical principle, wherein infinite potential is reduced from a mathematical luxury to a thermodynamic transaction. That which cannot be admitted by the system is not covertly stashed away; it is extinguished, and this reduction of freedom leaves an indelible, thermodynamic scar upon the physical world. This cost is rigorously quantified via Landauer's principle, which compels the system to settle the energetic account for every restriction of its degrees of freedom. When uninhibited, free dynamics is confronted with an uncompromising projection gate, non-viable dynamical freedom is brutally stripped away. This eliminated structure represents no passive absence, but an active selection residual that enforces unavoidable dissipation. The entropy balance permits no loopholes: the very work of constraining reality into a specific, permissible state consumes thermodynamic currency (ΔStotal = ΔSdyn + ΔSsel ≥ 0). Reality therefore emerges not as the aggregate of all that could exist in theory, but strictly as that which can afford to be chosen. At the quantum level, this mechanism is formalized through Projection-Constrained Schrödinger Dynamics (PCSE), where the evolution of the wave function is subjected to explicit structural constraint (iℏ ∂t ψ = Hψ − iℏλ(I − ΠK)ψ). Within the admissible subspace, unitarity and phase coherence are strictly preserved under Hamiltonian control. However, the instant intrinsic dynamics attempts to leak into forbidden configurations, it is countered by irreversible, dissipative damping. A fundamental, continuous tension arises between the coherent generation of possibilities and the system's unyielding demand for legality—a balance wherein any state violating the projection operator is exponentially driven to extinction. This uncompromising demand for viability fundamentally cleanses cosmology's most overcrowded landscape, specifically the unwieldy moduli space of string and M-theory vacua (> 10500). Rather than defaulting to the obscure premise that an infinity of parallel universes must co-exist to salvage mathematical consistency, the framework establishes an absolute boundary between theoretical solution space and physical realization. A physical universe cannot persist merely as an ephemeral equation; it must manifest as a fixed point capable of withstanding its own projection over time (x☆ ∈ Fix(ΠK ∘ Φ)). Only states anchored in the forward-invariant viability kernel are granted physical form. The remainder remains strictly solution space—barred from existence by a universe that never ceases to demand thermodynamic coverage for all that is called real. Core Theoretic & Quantitative Modules: Thermodynamic Selection & Minimal Landauer Bound: Free dynamics x't+1 = Φ(xt) generates candidate states, while realized dynamics xt+1 = ΠK(x't+1) is enforced by an idempotent projection operator. The selection residual ΔK(xt) = Φ(xt) − ΠK(Φ(xt)) represents discarded dynamical freedom. Whenever ΔK ≠ 0, N bits of physically encoded structure are erased, imposing the dissipative lower bound Qmin ≥ N kB T ln 2 and non-negative entropy production ΔStotal = ΔSdyn + ΔSsel ≥ 0. Hamiltonian Energy Formulation: Effective system energy HΠK = (χ/2) dF2(Kγ, Kref) + BA + (ζ/2) |OL(γ)|2 + Hirr attains its absolute minimum when kernel, barrier, witness, and irreversibility conditions are satisfied simultaneously: Esel = (χ/2) ||OK||2 + BA + Hirr. Projection-Constrained Schrödinger Dynamics (PCSE): iℏ ∂t ψ = Hψ − iℏλ(I − ΠK)ψ. Hamiltonian and unitary within admissible subspace K; illegal components exterior to K are damped non-unitarily at rate λ. For invariant subspaces ([ΠK, H] = 0), exponential convergence is exact: ||q(t)||2 = e−2λt ||q(0)||2. When [ΠK, H] ≠ 0, a dynamic equilibrium balances generation and dissipation. Vacuum Admissibility in String/M-Theory: On moduli space M, the cardinal distinction holds: Realization ≠ Solution; Realization ⊆ Solution Space. Physical vacua are selected as fixed points under forward-invariant viability: x☆ ∈ Fix(ΠK ∘ Φ) ∩ Γphys, where Γphys = { x0 ∈ M | Φt(x0) ∈ K ∀ t ≥ 0 }. Conventional unconstrained dynamics is recovered in the zero-projection limit (OK = 0). Epistemic Demarcation (AGENTS.md): Verified via L1 deterministic Python reference engine (thermodynamic_quantum_selection_engine.py) with 10/10 unit tests PASS (0.002s) under the cardinal doctrine INSPIRED_BY != IMPLEMENTS != PROVES.