Quantum, Classical and Symmetry Aspects of Max Born Reciprocity

Quantum, Classical and Symmetry Aspects of Max Born Reciprocity L.M. Tomilchik B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus Minsk, 16–19 May 2017 1 / 36 Topics 1 Reciprocal Symmetry and Maximum Tension Principle 2 Maximum Force and Newton gravity 3 Extended Phase Space (QTPH) as a Basic Manifold 4 Complex Lorentz group with Real Metric as Group of Reciprocal Symmetry 5 One-Particle Quasi-Newtonian Reciprocal-Invariant Hamiltonian dynamics 6 Canonic Quantization: Dirac Oscillator as Model of Fermion with a Plank mass 2 / 36 Born Reciprocity version 1 Reciprocity Transformations (RT): <ul style="display: flex;"><li style="flex:1">xµ </li><li style="flex:1">pµ </li><li style="flex:1">pµ </li><li style="flex:1">xµ </li></ul>qe →,→ − .<ul style="display: flex;"><li style="flex:1">qe </li><li style="flex:1">pe </li><li style="flex:1">pe </li></ul>2 Lorentz and Reciprocal Invariant quadratic form: 1qe2 1pe2 SB2 =xµxµ + pµpµ, ηµν = diag{1, −1, −1, −1} 3 xµ, pµ are quantum-mechanical canonic operators: [xµ, pν] = i~ηµν 4 Phenomenological parameter according definition pe = Mc so that qe = ~/Mc = Λc (Compton length). The mass M is a free model parameter. 5 Two dimensional constants qe[length], pe[momentum] connected by corellation: qepe = ~ (Plank constant). 3 / 36 Relativistic Oscillator Equation (ROE) Selfreciprocal-Invariant Quantum mechanical Equation: <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>2∂−+ ξµξµ Ψ(ξ) = λBΨ(ξ), ∂ξµ∂ξµ where ξµ = xµ/Λc. The (ROE) symmetry ≈ U(3, 1). There are solution corresponding linear discrete mass spectrum. 4 / 36 Maximum Tension Principle (MTP) The space-time and momentum variables are to be just the same dimentions of a quantity. It is necessary to have the universal constant with dimention: momentum/length, or equivalently: mass/time, energy/time, momentum/time. In reality: MTP [Gibbons, 2002] Gibbons Limit: ꢀꢀ ꢁdp dt c4 Maximum force Fmax ==<ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">= FG, </li></ul>4GN max ꢁdE c5 Maximum power Pmax <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">= PG. </li></ul>dt 4GN max [momentum]/[length] – Universal constant pe qe c3 <ul style="display: flex;"><li style="flex:1">æ0 = </li><li style="flex:1">=</li></ul>[LMT, 1974, 2003] 4GN <ul style="display: flex;"><li style="flex:1">−1 </li><li style="flex:1">−2 </li></ul>It is evident: æ0 = c F= c P. All these values are <ul style="display: flex;"><li style="flex:1">max </li><li style="flex:1">max </li></ul>essentially classic (does not contain Plank constant ~). 5 / 36 Born Reciprocity and Gibbons Limit Let the parameter a to be some fixed (nonzero!) value of action S. We replace initial Born’s relation qepe = ~ by πqepe = a without beforehand action discretness supposition. We demand the nonzero area Sfix = πqepe of the phase plane only. The second suggestion is: pe qe c3 <ul style="display: flex;"><li style="flex:1">æ0 = </li><li style="flex:1">=</li></ul>(universal constant). 4GN 6 / 36 Now the parameters qe, pe are defined separately <ul style="display: flex;"><li style="flex:1">r</li><li style="flex:1">r</li></ul>aaæ qe = ,pe = πæ πEvidently: the phenomenological constants Ee = pec [Energy] and te = c−1qe [time] satisfy the conditions Ee te c5 Eete = πa, = c2æ = 4GN Under Born’s parametrization 2MGN pe = Mc, qe = = rg(M) – gravitational radius. c2 General mass-action correlation ac . 4πGN M2 = It is valid as in classical as in quantum case. Under supposition when amin = h we receive shc 12~c GN M2 = =MP2 l ,MP l =– Plank mass. 4πGN 7 / 36 Maximum ꢂFꢂorce and Modified Newton Gravity mM r2 c4 2mGN 2MGN 1r2 ꢂ ꢂ (NG): f = GN ≡4GN <ul style="display: flex;"><li style="flex:1">c2 </li><li style="flex:1">c2 </li></ul>prgRg r2 r02 = FG ≡ FG (r > 0, r0 = rgRg) r2 r02 G r2 ꢂꢂꢂ<ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢂ</li></ul>ꢂꢂ ꢂꢂꢂmax ꢂ(ModNG): fmod = F (r > r0), fmod = FG ꢂ<ul style="display: flex;"><li style="flex:1">N</li><li style="flex:1">N</li></ul>r=r0 The corresponding gravitational potential Energy r02 mod gr U= −FG rpossess a minimum under r = r0 c4 2GN c2 <ul style="display: flex;"><li style="flex:1">√</li><li style="flex:1">√</li></ul>Ugmrod(min) = −FGr0 = − mM = − mM 4GN c2 2We have dealing (instead of point-like masses) with a pair spacely extended simple massive gravitating objects like the elastic balls or drops with a high surface tension. 8 / 36 r0 <ul style="display: flex;"><li style="flex:1">M</li><li style="flex:1">m</li></ul><ul style="display: flex;"><li style="flex:1">Rg </li><li style="flex:1">rg </li></ul>√12Erest(M, m) = Mc2 + mc2 − mMc2 9 / 36 The whole energy such a “binary” system is as follows: √1E(M, m) = Mc2 + mc2 − mMc2 2ꢃr ꢄ12µ= (M + m)c2 1 − , M + m where µ = mM/(m + M). Mc2, mc2, (M + m)c2 – rest energy p√12 12<ul style="display: flex;"><li style="flex:1">∆E = </li><li style="flex:1">µ(M + m)c2 = </li></ul>mMc2 – energy corresponding √12the “mass defect” ∆m = mM 10 / 36 “Gravitational fusion” picture [eliminated energy] ∆E (M + m)c2 λ = =1[rest energy] <ul style="display: flex;"><li style="flex:1">√</li><li style="flex:1">√</li></ul><ul style="display: flex;"><li style="flex:1">mM </li><li style="flex:1">δ</li></ul><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">=</li></ul>,<ul style="display: flex;"><li style="flex:1">2(M + m) </li><li style="flex:1">2 1 + δ </li></ul>where mMmin Mmax <ul style="display: flex;"><li style="flex:1">δ = </li><li style="flex:1">,</li></ul>δ0 6 δ 6 1, δ0 = M∆Eeliminated 14Maximum λmax <ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">=</li></ul>Erest under δ = 1 (m = M) Such a “fusion” of two equal rest mass M lead to mass = 3M/2, the Energy ∆E = Mc2/2 eliminate (3 : 1). 11 / 36 Extended Phase Space as a Basic Manifold Space of states (QTP) and Energy (H) (sec [J. Synge, 1962]) in general: 2 + 2N dimensions. The metric is N<ul style="display: flex;"><li style="flex:1">}| </li><li style="flex:1">z</li><li style="flex:1">{</li></ul>ηAB = diag{1, −1, −1, . . . , −1} (X, Y ) – pair of conjugated pseudoeuclidean vectors XA, YA, A, B = 0, 1, . . . , N (N + 1 dimention). The Poisson brackets for ϕ(X, Y ), f(X, Y ), <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>NX∂ϕ ∂f ∂Xk ∂Yk ∂ϕ ∂f ∂Y k ∂Xk {ϕ, f}A,B =−k=0 so that {XA, YA}A,B = ηAB 12 / 36 Dynamic System in (QT, P H) is defined, when the Energy Surface (2N + 1 dimention) Ω(X, Y ) = 0 is defined. Solving this equation on Y0 we obtain Energy equation Y0 = F (Xk, Yl, X0) k, l = 1, 2, . . . , N The corresponding Hamilton function of the System is by def: H ≡ Y0 = H (Xk, Yl, Y0) We shall consider (2 + 2 · 3) – dimensional version (QT, P H) A, B → µ, ν = 0, 1, 2, 3, ηµν = diag{1, −1, −1, −1}, Xµ, Xµ are real 4-dimensional pseudoeuclidean SO(3, 1) – vectors. 13 / 36 Complexification of (QT, P H) Let us introduce the complex 4-vector Zµ according definition: Zµ = Xµ + iY µ, ηµν = diag{1, −1, −1, −1} and define the real norm ZµZµ∗ = ZµηµνZµ∗ <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">2</li></ul>= |Z0| − |Z1| − |Z2| − |Z3| = inv 14 / 36 Complex Lorentz Group with a real metric (Barut Group [Barut 1964]) The group of 4 × 4 complex matrices Λ of transformations Z0 = ΛZ, Z = X + iY satisfying the condition ΛηΛ† = η, († – hermitien conjugation) ¯Notice: the diagonal matrices ΛR and ΛR ∈ (BG) ΛR = −iI4 = −idiag{1, 1, 1, 1} and <ul style="display: flex;"><li style="flex:1">¯</li><li style="flex:1">ΛR = −iηµν = −idiag{1, −1, −1, −1} </li></ul>Metric invariant (no positive-defined) <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">2</li></ul>ZµZµ∗ = |Z0| − |Z1| − |Z2| − |Z3| = inv 22Three possibilities: |Z| ≷ 0, |Z| = 0 – isotropic case 15 / 36 SU(3, 1) – subgroup structure (QTPH) SOvHQT = {r, ct} = xµ Lk ꢅP H = p, E/c = pµ ~βv = ~v/c Ccenter *C4 SOHUkl ‡ SUH(PTQH) ꢅP T = p, F0t = Πµ Mk QH = {r, E/F0} = Ξµ SOf H<ul style="display: flex;"><li style="flex:1">~</li><li style="flex:1">~</li></ul>βf = f/F0 F0 = Fmax k, l = 1, 2, 3 16 / 36 Reciprocal Invariant SB2 as BG metric invariant ZµZµ∗ By def: Zµ = Xµ + iYµ <ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">1</li></ul><ul style="display: flex;"><li style="flex:1">Xµ = </li><li style="flex:1">xµ = </li></ul>{x0, xk}, Yµ = pµ = {p0, pk}. <ul style="display: flex;"><li style="flex:1">qe </li><li style="flex:1">qe </li><li style="flex:1">pe </li><li style="flex:1">pe </li></ul>Then: <ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul>ꢁ<ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul><ul style="display: flex;"><li style="flex:1">3</li><li style="flex:1">3</li></ul><ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul>1qe2 1pe2 SB2 =x20 − xk2 +p20 − pk2 =<ul style="display: flex;"><li style="flex:1">k=1 </li><li style="flex:1">k=1 </li></ul>ꢀ<ul style="display: flex;"><li style="flex:1">3</li><li style="flex:1">3</li></ul>x02 qe2 p02 pe2 xk2 qe2 pk2 pe2 <ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">+</li></ul>−+<ul style="display: flex;"><li style="flex:1">= |Z0| − </li><li style="flex:1">|Zk| = ZµZµ∗ </li></ul><ul style="display: flex;"><li style="flex:1">k=1 </li><li style="flex:1">k=1 </li></ul>Let xµ, pµ to be 4-vectors under SOv(3, 1)-transformations 17 / 36 Notice! There exist alternative: ¯Zµ = Xµ + iYµ <ul style="display: flex;"><li style="flex:1">¯</li><li style="flex:1">¯</li></ul>where <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul><ul style="display: flex;"><li style="flex:1">x0 pk </li><li style="flex:1">p0 xk </li></ul>,pe qe def =def =<ul style="display: flex;"><li style="flex:1">¯</li><li style="flex:1">¯</li></ul>

Quantum, Classical and Symmetry Aspects of Max Born Reciprocity

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