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CRC Press eBooks

Perturbation Theory for Scattering States in One Dimension

Francisco Marcelo Fernández

Published Sep 19, 2000• 0 citations
Publisher

Abstract

We can view equations (8.1) and (8.2) as a system of two linear equations with two unknowns c1 and c2. The determinant of such a system is the Wronskian W(u, v) = uv′ − u′v . (8.3) If W(u, v) is nonzero for some value of x, then the only solution of the system of homogeneous linear equations is c1 = c2 = 0, and the functions are linearly independent. If, on the other hand, W(u, v) is zero for all values of x, then there are solutions for nonzero values of c1 and c2, and the functions are linearly dependent.

DOI
10.1201/9781420039641-11

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Stationary State Perturbation Theory

Kurt Gottfried

The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.

CRC Press eBooks•Mar 9, 2018•0 citations

Perturbation Theory for Scattering States in One Dimension

CRC Press eBooks•Sep 19, 2000•0 citations

Green's-Function Formalism of the One-Dimensional Heisenberg Spin System

Jun Kondo, Yamaji K.

The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.

Progress of Theoretical Physics•Mar 1, 1972•246 citations

Note on the General Theory of Scattering

M. L. Goldberger

The usual presentation of scattering theory proceeds from a discussion of the integral equations(1) ψa(±)=Φa+ 1/Ea±iε-H0 H1ψa(±).

Physical Review•Jun 1, 1951•4 citations

Independence, Algebra, And Action Compiling the Complex Scalar Field and Hamilton's Principle

Dean Kulik

Independence, Algebra, And Action Compiling the Complex Scalar Field and Hamilton's Principle Driven by Dean Kulik July 2026 Abstract Standard quantum mechanics assumes complex amplitudes. Classical and quantum mechanics assume Hamilton's principle. Neither is derived; both are imported. This paper compiles both from the same five primitive constraints {C0, C1, C1a, C2, C5} established in Paper 1. Two main theorems are proved. First, the Division Algebra Theorem: the unique admissible scalar continuation algebra is the complex numbers C, established by reducing to the Frobenius candidates {R, C, H}, eliminating R via C5's dual degree-of-freedom requirement, and eliminating H via the simplicity of H as a real algebra combined with C1a's independence constraint. Second, the Extremal Theorem: a field configuration is locally admissible under the compiled constraints if and only if it extremizes the action functional delta S = 0. Hamilton's principle is the global expression of local C1-consistency. A compiled Measurement Theorem follows from the channel decomposition forced by C5: collapse is a C2 history-update, not a C1 transition. The independence constraint [T_A, T_B] = 0 serves as the structural spine: it compiles conservation (C3 to L3), forces the complex scalar field (Division Algebra Theorem), and will constrain gauge sector factorization. One axiom. Three compiled results. §1 Introduction Two of the deepest assumptions in physics are that quantum amplitudes are complex-valued and that physical trajectories extremize the action. Both are usually treated as foundational inputs. The goal of this paper is to show both are compiled consequences of a primitive constraint set that mentions neither. The companion paper (Paper 1) compiled: conservation, locality, the free-field Lagrangian L, and the wave equation, from the five primitive constraints {C0, C1, C1a, C2, C5}. The present paper derives two further results from the same set, and establishes the structural connection between them. The two theorems share the same spine. The independence constraint C1a — that independent distinctions continue independently — has the algebraic form [T_A, T_B] = 0 for independent operators. This single condition eliminates quaternionic scalar algebras (Division Algebra Theorem), blocks coordinated creation-destruction channels (compiling conservation), and will factorize the gauge group (T5, T6). Section 7 develops this as the Independence Spine Theorem. Paper scope. Both theorems concern the free field (Paper 1, C8). Interactions require C9 (equivalence class coupling) and are addressed in Paper 4. The Measurement Theorem (Section 5) is fully compiled. The Born rule requires one additional input (C7 geometry; see Section 5.3 and open problem Omega_B). §2 The Primitive Constraints The irreducible primitive set established in Paper 1 is {C0, C1, C1a, C2, C5}. Conservation (L3) and locality (L4) are compiled consequences of this set, not primitives. The definitions are reproduced here for self-containment. C0 (Distinction exists): At least two states exist that are not identical. This is the minimum condition for any structure to be meaningful. C1 (Continuation exists): Every admissible state s admits at least one successor state s' with s' not equal to s under the transition algebra. Terminal states and stationary continuation are excluded. The identity element e of the composition algebra (empty composition, satisfying e composed with T = T) is required by the algebra but is not itself a physical continuation event (Lemma B, Section 4.2). C1a (Independence Preservation): Independent distinctions remain independently continuable. The continuation of A cannot depend on the presence or continuation of B when A and B are independent. C2 (History Preservation): Every continuation preserves the set of states reachable from prior states in the history. C2 forbids deletion of accessible states from the transition graph and establishes the ordering relation from reachability. C5 (Observable Event Primitive): An observable event is the minimum continuation that simultaneously carries two independent degrees of freedom: propagation (a continuous spatial datum) and residue (a distinct temporal datum). C5 bridges transition structure and observation, and forces dim_R(D) >= 2 for any admissible scalar algebra D (Lemma D, Section 4.2). §3 The Independence Constraint and Its Algebraic Form C1a is the primitive that does the most structural work in this paper. Its algebraic form is derived immediately. Let T_A and T_B be continuation operators acting on independent subsystems A and B. Their joint continuation must factor as T_AB = T_A ⊗ T_B. Factorization requires that the result is independent of which update was applied first. Otherwise the joint continuation depends on an ordering between independent events, which is additional structure not licensed by C0 to C2. Therefore: [T_A, T_B] = T_A T_B − T_B T_A = 0 for all independent A, B Non-commutativity of independent continuation operators is a direct C1a violation by definition, not a violation of an algebraic nicety. This single equation is the structural spine of the paper. Scope. This commutation requirement applies to the scalar continuation algebra — the algebra of weights on independent continuation amplitudes. Physical symmetry operators (angular momentum generators, Lorentz boosts, gauge generators) act at a different layer and are not required to commute. This restriction is explicit and essential throughout. §4 The Division Algebra Theorem §4.1 Frobenius Reduction [C] Theorem [C]. The admissible scalar continuation algebra D is a finite-dimensional division algebra over R. Proof. The compiled linear state space (C1 + C1a, Paper 1 §4.1) is a vector space over some scalar field D. Four properties of D follow from the primitives: No zero divisors (from C0 + C1): C0 requires nonzero states to remain distinguishable under scalar multiplication. If scalar d ≠ 0 maps nonzero state ψ to zero, two distinct states become indistinguishable — a C0 violation. Additionally, C1 requires that every continuation produces a nonzero successor amplitude. If two nonzero amplitudes a and b satisfy a·b = 0 in D, then applying amplitude a followed by b to a lawful continuation produces a terminated (zero) amplitude state — a C1 violation. Therefore D has no zero divisors from both C0 and C1. Unit element (from C1): C1 requires the composition algebra to have an empty composition (Lemma B). This is the unit element of D. Division algebra (from no zero divisors + unit + finite dimension): For finite-dimensional associative algebras over a field, the combination of no zero divisors and a unit element is equivalent to being a division algebra. Every nonzero element has a two-sided inverse. Over R and finite-dimensional (from C5 + M1): Real scaling is the minimal requirement for continuous amplitude variation. C5 forces dim_R(D) ≥ 2 (see Lemma D). The minimality principle (M1) selects the minimal structure with no unconstrained degrees of freedom: dim_R(D) = exactly 2. Infinite-dimensional scalar structures carry unconstrained degrees of freedom not licensed by any primitive and are excluded by M1. Frobenius (1877): The only finite-dimensional division algebras over R are R (dimension 1), C (dimension 2), and H (dimension 4). Therefore D ∈ {R, C, H}. □ §4.2 The Four Admissible Conditions Any admissible scalar continuation algebra must satisfy four conditions, each compiled from the primitives. Together with Frobenius, these identify C as the unique survivor. Lemma A (Associativity) [C] A physical continuation history is an ordered sequence of lawful continuations. Any algebra representing that history must assign the same aggregate continuation map regardless of how the sequence is parenthesized. If associativity failed, the parenthesization — a bookkeeping convention — rather than the physical history would determine the outcome, violating C0. Therefore D is associative. □ Lemma B (Identity) [C] Sequential composition requires a neutral element — the empty composition e satisfying e ∘ T = T ∘ e = T for all T. The empty composition is not a continuation event; it represents the absence of composition, a structural requirement of the algebra. C1 excludes stationary continuation (T(s) = s as a physical event). It does not exclude the neutral element required by the composition algebra. These apply at different layers: e is required by the algebra and excluded as a physical continuation by C1. No contradiction. Identity is correctly placed in the representational layer. □ Lemma C (Commutativity of the scalar algebra) [C] The scalar weight on a continuation amplitude is itself a continuation operation ψ ↦ a·ψ. For independent amplitudes ψ_A and ψ_B with independent weights a_A and a_B, applying the weights in either order must give the same result — otherwise the combined amplitude depends on ordering between independent operations, which is a direct C1a violation. Therefore a_A · a_B = a_B · a_A for all elements of D. D is commutative. □ Lemma D (Two-dimensional) [C + M1] C5 requires every observable event to simultaneously carry two independent degrees of freedom: propagation and residue. These are not the same degree of freedom. A one-dimensional algebra has only one real degree of freedom and its only intrinsic transformation generates Z_2, which is discrete and cannot encode continuous propagation. Therefore: C5 forces: dim_R(D) ≥ 2 (lower bound) M1 selects: dim_R(D) = 2 (minimality) Higher-dimensional structures (quaternions, octonions, and beyond) carry additional degrees of freedom not licensed by any primitive. They are excluded by M1, not by a physical prohibition. The paper answers 'why not four?' with: four is not forced. Two is forced, and M1 selects the minimum forced structure. □ §4.3

•Jul 26, 2026•0 citations

Information, Quantum Probability and Reflection/Refraction Part II Dynamic Probability

Francesco R. Ruggeri

In Part I we considered light moving in the positive x direction stiking a medium with a different index of refraction with some reflecting and the rest refracting. We argued one may immediately write a conservation of energy equation, but this is a single equation while there are two unknowns, the intensity of the reflected beam and the refracted, assuming the incident intensity is known. (We also argued that one also has the relation E=pc for a photon, but this is not needed to solve for the two intensities.) Our main point was that instead of seeking another independent equation (containing new information), one may write the conservation of energy equation as a product of two equations A+ B=C ((1a)) and A/c1 -B/c1 = C/c2 ((1b)) where AA/c1 = BB/c1 + CC/c2 is the energy conservation equation. Thus energy conservation contains all of the information needed, but one must move to a square root of probability picture in order to obtain the two equations needed to solve for the reflected and refracted values B,C and then the intensities BB and CC. In Part I, we noted that the reflection/refraction problem may be presented as a steady state one in which the scenario is not changing in time. A Maxwell-Boltzmann gas in equilibrium also does not change in time, but here there is a difference, namely a lack of time reversal invariance. In other words there is a distinct direction of motion i.e. the incident and refracted beams move in the positive direction only and the reflected in the negative x. The problem, however, is also a probabilistic one because given a single photon, it either reflects or refracts with certain probabilities. Thus we suggest there exists in this physical steady state problem a dynamic or directional probability. This probability is related to space and may be positive or negative unlike the usual time probabilities linked to events i.e. classical probability theory. The conservation of energy equation is a classical probability one, but dynamic motion probabilities are present in the problem and should also be manifest in equations. In fact the two dynamic probability equations allow for the solving of the reflected/refracted intensities (probabilities), but are also directly linked to the conservation of energy equation which may be written as: AA/c1 -BB/c1 = CC/c2 which points towards flow in two different directions. Thus we argue that a dynamic directional probability is responsible for the decomposition of the energy conservation equation into two dynamic probability ones.

•Dec 7, 2022•0 citations
We can view equations (8.1) and (8.2) as a system of two linear equations with two unknowns c1 and c2. The determinant of such a system is the Wronskian W(u, v) = uv′ − u′v . (8.3) If W(u, v) is nonzero for some value of x, then the only solution of the system of homogeneous linear equations is c1 = c2 = 0, and the functions are linearly independent. If, on the other hand, W(u, v) is zero for all values of x, then there are solutions for nonzero values of c1 and c2, and the functions are linearly dependent.
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
Independence, Algebra, And Action Compiling the Complex Scalar Field and Hamilton's Principle Driven by Dean Kulik July 2026 Abstract Standard quantum mechanics assumes complex amplitudes. Classical and quantum mechanics assume Hamilton's principle. Neither is derived; both are imported. This paper compiles both from the same five primitive constraints {C0, C1, C1a, C2, C5} established in Paper 1. Two main theorems are proved. First, the Division Algebra Theorem: the unique admissible scalar continuation algebra is the complex numbers C, established by reducing to the Frobenius candidates {R, C, H}, eliminating R via C5's dual degree-of-freedom requirement, and eliminating H via the simplicity of H as a real algebra combined with C1a's independence constraint. Second, the Extremal Theorem: a field configuration is locally admissible under the compiled constraints if and only if it extremizes the action functional delta S = 0. Hamilton's principle is the global expression of local C1-consistency. A compiled Measurement Theorem follows from the channel decomposition forced by C5: collapse is a C2 history-update, not a C1 transition. The independence constraint [T_A, T_B] = 0 serves as the structural spine: it compiles conservation (C3 to L3), forces the complex scalar field (Division Algebra Theorem), and will constrain gauge sector factorization. One axiom. Three compiled results. §1 Introduction Two of the deepest assumptions in physics are that quantum amplitudes are complex-valued and that physical trajectories extremize the action. Both are usually treated as foundational inputs. The goal of this paper is to show both are compiled consequences of a primitive constraint set that mentions neither. The companion paper (Paper 1) compiled: conservation, locality, the free-field Lagrangian L, and the wave equation, from the five primitive constraints {C0, C1, C1a, C2, C5}. The present paper derives two further results from the same set, and establishes the structural connection between them. The two theorems share the same spine. The independence constraint C1a — that independent distinctions continue independently — has the algebraic form [T_A, T_B] = 0 for independent operators. This single condition eliminates quaternionic scalar algebras (Division Algebra Theorem), blocks coordinated creation-destruction channels (compiling conservation), and will factorize the gauge group (T5, T6). Section 7 develops this as the Independence Spine Theorem. Paper scope. Both theorems concern the free field (Paper 1, C8). Interactions require C9 (equivalence class coupling) and are addressed in Paper 4. The Measurement Theorem (Section 5) is fully compiled. The Born rule requires one additional input (C7 geometry; see Section 5.3 and open problem Omega_B). §2 The Primitive Constraints The irreducible primitive set established in Paper 1 is {C0, C1, C1a, C2, C5}. Conservation (L3) and locality (L4) are compiled consequences of this set, not primitives. The definitions are reproduced here for self-containment. C0 (Distinction exists): At least two states exist that are not identical. This is the minimum condition for any structure to be meaningful. C1 (Continuation exists): Every admissible state s admits at least one successor state s' with s' not equal to s under the transition algebra. Terminal states and stationary continuation are excluded. The identity element e of the composition algebra (empty composition, satisfying e composed with T = T) is required by the algebra but is not itself a physical continuation event (Lemma B, Section 4.2). C1a (Independence Preservation): Independent distinctions remain independently continuable. The continuation of A cannot depend on the presence or continuation of B when A and B are independent. C2 (History Preservation): Every continuation preserves the set of states reachable from prior states in the history. C2 forbids deletion of accessible states from the transition graph and establishes the ordering relation from reachability. C5 (Observable Event Primitive): An observable event is the minimum continuation that simultaneously carries two independent degrees of freedom: propagation (a continuous spatial datum) and residue (a distinct temporal datum). C5 bridges transition structure and observation, and forces dim_R(D) >= 2 for any admissible scalar algebra D (Lemma D, Section 4.2). §3 The Independence Constraint and Its Algebraic Form C1a is the primitive that does the most structural work in this paper. Its algebraic form is derived immediately. Let T_A and T_B be continuation operators acting on independent subsystems A and B. Their joint continuation must factor as T_AB = T_A ⊗ T_B. Factorization requires that the result is independent of which update was applied first. Otherwise the joint continuation depends on an ordering between independent events, which is additional structure not licensed by C0 to C2. Therefore: [T_A, T_B] = T_A T_B − T_B T_A = 0 for all independent A, B Non-commutativity of independent continuation operators is a direct C1a violation by definition, not a violation of an algebraic nicety. This single equation is the structural spine of the paper. Scope. This commutation requirement applies to the scalar continuation algebra — the algebra of weights on independent continuation amplitudes. Physical symmetry operators (angular momentum generators, Lorentz boosts, gauge generators) act at a different layer and are not required to commute. This restriction is explicit and essential throughout. §4 The Division Algebra Theorem §4.1 Frobenius Reduction [C] Theorem [C]. The admissible scalar continuation algebra D is a finite-dimensional division algebra over R. Proof. The compiled linear state space (C1 + C1a, Paper 1 §4.1) is a vector space over some scalar field D. Four properties of D follow from the primitives: No zero divisors (from C0 + C1): C0 requires nonzero states to remain distinguishable under scalar multiplication. If scalar d ≠ 0 maps nonzero state ψ to zero, two distinct states become indistinguishable — a C0 violation. Additionally, C1 requires that every continuation produces a nonzero successor amplitude. If two nonzero amplitudes a and b satisfy a·b = 0 in D, then applying amplitude a followed by b to a lawful continuation produces a terminated (zero) amplitude state — a C1 violation. Therefore D has no zero divisors from both C0 and C1. Unit element (from C1): C1 requires the composition algebra to have an empty composition (Lemma B). This is the unit element of D. Division algebra (from no zero divisors + unit + finite dimension): For finite-dimensional associative algebras over a field, the combination of no zero divisors and a unit element is equivalent to being a division algebra. Every nonzero element has a two-sided inverse. Over R and finite-dimensional (from C5 + M1): Real scaling is the minimal requirement for continuous amplitude variation. C5 forces dim_R(D) ≥ 2 (see Lemma D). The minimality principle (M1) selects the minimal structure with no unconstrained degrees of freedom: dim_R(D) = exactly 2. Infinite-dimensional scalar structures carry unconstrained degrees of freedom not licensed by any primitive and are excluded by M1. Frobenius (1877): The only finite-dimensional division algebras over R are R (dimension 1), C (dimension 2), and H (dimension 4). Therefore D ∈ {R, C, H}. □ §4.2 The Four Admissible Conditions Any admissible scalar continuation algebra must satisfy four conditions, each compiled from the primitives. Together with Frobenius, these identify C as the unique survivor. Lemma A (Associativity) [C] A physical continuation history is an ordered sequence of lawful continuations. Any algebra representing that history must assign the same aggregate continuation map regardless of how the sequence is parenthesized. If associativity failed, the parenthesization — a bookkeeping convention — rather than the physical history would determine the outcome, violating C0. Therefore D is associative. □ Lemma B (Identity) [C] Sequential composition requires a neutral element — the empty composition e satisfying e ∘ T = T ∘ e = T for all T. The empty composition is not a continuation event; it represents the absence of composition, a structural requirement of the algebra. C1 excludes stationary continuation (T(s) = s as a physical event). It does not exclude the neutral element required by the composition algebra. These apply at different layers: e is required by the algebra and excluded as a physical continuation by C1. No contradiction. Identity is correctly placed in the representational layer. □ Lemma C (Commutativity of the scalar algebra) [C] The scalar weight on a continuation amplitude is itself a continuation operation ψ ↦ a·ψ. For independent amplitudes ψ_A and ψ_B with independent weights a_A and a_B, applying the weights in either order must give the same result — otherwise the combined amplitude depends on ordering between independent operations, which is a direct C1a violation. Therefore a_A · a_B = a_B · a_A for all elements of D. D is commutative. □ Lemma D (Two-dimensional) [C + M1] C5 requires every observable event to simultaneously carry two independent degrees of freedom: propagation and residue. These are not the same degree of freedom. A one-dimensional algebra has only one real degree of freedom and its only intrinsic transformation generates Z_2, which is discrete and cannot encode continuous propagation. Therefore: C5 forces: dim_R(D) ≥ 2 (lower bound) M1 selects: dim_R(D) = 2 (minimality) Higher-dimensional structures (quaternions, octonions, and beyond) carry additional degrees of freedom not licensed by any primitive. They are excluded by M1, not by a physical prohibition. The paper answers 'why not four?' with: four is not forced. Two is forced, and M1 selects the minimum forced structure. □ §4.3
In Part I we considered light moving in the positive x direction stiking a medium with a different index of refraction with some reflecting and the rest refracting. We argued one may immediately write a conservation of energy equation, but this is a single equation while there are two unknowns, the intensity of the reflected beam and the refracted, assuming the incident intensity is known. (We also argued that one also has the relation E=pc for a photon, but this is not needed to solve for the two intensities.) Our main point was that instead of seeking another independent equation (containing new information), one may write the conservation of energy equation as a product of two equations A+ B=C ((1a)) and A/c1 -B/c1 = C/c2 ((1b)) where AA/c1 = BB/c1 + CC/c2 is the energy conservation equation. Thus energy conservation contains all of the information needed, but one must move to a square root of probability picture in order to obtain the two equations needed to solve for the reflected and refracted values B,C and then the intensities BB and CC. In Part I, we noted that the reflection/refraction problem may be presented as a steady state one in which the scenario is not changing in time. A Maxwell-Boltzmann gas in equilibrium also does not change in time, but here there is a difference, namely a lack of time reversal invariance. In other words there is a distinct direction of motion i.e. the incident and refracted beams move in the positive direction only and the reflected in the negative x. The problem, however, is also a probabilistic one because given a single photon, it either reflects or refracts with certain probabilities. Thus we suggest there exists in this physical steady state problem a dynamic or directional probability. This probability is related to space and may be positive or negative unlike the usual time probabilities linked to events i.e. classical probability theory. The conservation of energy equation is a classical probability one, but dynamic motion probabilities are present in the problem and should also be manifest in equations. In fact the two dynamic probability equations allow for the solving of the reflected/refracted intensities (probabilities), but are also directly linked to the conservation of energy equation which may be written as: AA/c1 -BB/c1 = CC/c2 which points towards flow in two different directions. Thus we argue that a dynamic directional probability is responsible for the decomposition of the energy conservation equation into two dynamic probability ones.