QSCP XIII 2008 East Lansing A Quantum Mechanical Description of the Laws of Relativity • General formulation of Q.M. f – Resonances ö – Emergence of Jordan blocks • Complex Systems r – ODLRO b – Coherent Dissipative Systems i • Application to the Theory of Relativity l – Klein-Gordon-Dirac equations – Electromagnetic- and gravitational effects d – Schwarzschild gauge
QSCP XIII 2008 East Lansing Dilatation Analytic Scaling Balslev-Combes, Commun. Math. Phys 22 (1971) 280-294 D ( H ) = F Œ H , H F Œ H { } f ö U( J )= exp(iA J ) r k = N r r k + r r b  A= 1 p x x p [ ] k k k 2 i k = 1 l d
QSCP XIII 2008 East Lansing U is a unitary one-parameter dilation group realizing the scaling below. Continuing q to complex values defines under certain conditions a dilatation analytic family of operators subject to the B-C theorem f H Æ UHU - 1 = H ( J ) ö r U ( J ) F (x 1 ,...,x N )= exp(iA J ) F (x 1 ,...,x N ) b 2 J ) F (e J x 1 ,...,e J x N ) = exp( 3N i l d
QSCP XIII 2008 East Lansing Assuming that the interaction is a sum of two-body ∆-compact operators and T the kinetic energy operator then H below has a compact analytic extension to the complex plane Ω. The B-C theorem concerns the spectrum of H e J Æ h = h e i J ; U ( J ) Æ U ( i J ) Æ U ( h ) f ö H = T + V Æ H ( h ) + V ( h ) = h - 2 T + V ( h ) r b H ( h ) F ( h ) = eF ( h ); h Œ W i l d
QSCP XIII 2008 East Lansing
QSCP XIII 2008 East Lansing
QSCP XIII 2008 East Lansing
QSCP XIII 2008 East Lansing • Här är plats för den första punkten • Här finns plats för den andra Plats för bild • Och den tredje osv.
QSCP XIII 2008 East Lansing Complex symmetric interaction • Analytic continuation of quantum mechanics • Complex scaling of unitary dilation group • Extended Hamiltonian spectrum • Dynamics - time evolution Ú Ú ( ) ( ) f r ( ) dr = ( ) ( ) f r ' ( ) dr ' j * r W r j * r '* W r ' bild 1 , r 2 ,... r N ; r ' = h 3 N r ; h = e i q ; J £ p / 2 r = r
QSCP XIII 2008 East Lansing ( a *) H ( a ) matrix elements analytic in a Y Y i j ( a *) ( a ) Y Y = d i j ij H ( a ) H ( a ) E ( a ) 0 Ê ˆ Ê ˆ 11 12 1 Á ˜ Á ˜ Æ Á ˜ Á ˜ H ( a ) H ( a ) 0 E ( a ) Ë ¯ Ë ¯ 21 22 2 Is this similarity transfor mation always possible?
CMMSE 200 CMMSE 2008 Murcia 8 Murcia QSCP XIII 2008 East Lansing Example : H 12 = H 21 = i u ; u Œ ¬ ; Ê Ê ˆ ˆ Ê Ê ˆ ˆ H 11 - i u Æ E 2 u Á Á ˜ ˜ Á Á ˜ ˜ - i u H 22 0 E Ë Ë ¯ ¯ Ë Ë ¯ ¯ With degenerate eigenvalues (H 11 - H 22 ) 2 - 4 u 2 = l ± = 1 2 (H 11 + H 22 ) ± l + = l - = 1 2 (H 11 + H 22 ) = H 11 ± u = E if H 22 = H 11 ± 2 u 1. Crossing rule. 2. Eigenvalue real. 3. Appearance of unphysical solutions
QSCP XIII 2008 East Lansing Coulomb Reconstruction
QSCP XIII 2008 East Lansing Unitary transformation between the real and the complex symmetric form i p m ( k + l - 2 ) Q = B - 1 JB; Q kl = ( d kl - 1 m ) e ; k , l = 1 , 2 ... m Ê Ê ˆ ˆ 0 1 0 . 0 Á Á ˜ ˜ 0 0 1 . . Á Á ˜ ˜ J = Á Á ˜ ˜ . . . . 0 . ; Á Á ˜ ˜ . . . . 1 . Á Á ˜ ˜ Á Á ˜ ˜ 0 . 0 . 0 Ë Ë ¯ ¯ m - 1 h Q h = h B - 1 QB h = f J f = Â f k f k + 1 k = 1
QSCP XIII 2008 East Lansing Ê Ê ˆ ˆ w 2 w m - 1 1 . w Á Á ˜ ˜ w 3 w 6 w 3 ( m - 1 ) 1 . Á Á ˜ ˜ i p 1 B = Á Á ˜ ˜ m ; w = e . . . . . m Á Á ˜ ˜ . . . . . Á Á ˜ ˜ Á Á ˜ ˜ 1 w 2m - 1 w 2 ( 2m - 1 ) . w ( m - 1 )( 2m - 1 ) Ë Ë ¯ ¯ h = h 1 ,h 2 ,...h m h B = g = g 1 ,g 2 ,...g m h B - 1 = f = f 1 , f 2 ,...f m
CMMSE 2008 Murcia QSCP XIII 2008 East Lansing i p r m ( k + l - 2 ) r = ( d kl - ( R r ) kl ) e Q ; k , l = 1 , 2 ... m ; r £ m kl Ï Ï sin ( p r ( l - k ) ) 1 m k ≠ l Ô Ô Ô Ô sin ( p ( l - k ) m ) ( R r ) kl = . Ì Ì m r Ô Ô m k = l Ô Ô Ó Ó
QSCP XIII 2008 East Lansing QUANTUM TECHNOLOGY Application Area Mechanism/Technique Condensed Matter Broken Symmetry Superconductivity SC ODLRO Quantum Hall Effect Topology High Temperature SC Quantum Statistics Superfluidity Gauge Symmetry SQUID Josephson Effect
QSCP XIII 2008 East Lansing Liouville von Neuman Equation: i h ∂ r ( t ) = ˆ L r ( t ) ∂ t ˆ r = A r + B r ; exp( ˆ ) = e A r e B P P P = - i ˆ ˆ t ; A = - iHt = B † L Bloch equation - ∂r ∂b = ˆ L B ˆ 2 ˆ P = - b B ; A = - b L 2 H = B
QSCP XIII 2008 East Lansing G = Y ( x 1 ... x N Y ( x 1 ... x N ; Tr{ G } = 1; G 2 = G = G † ' ) = G ( p ) ( x 1 ... x p x 1 ' ... x p Ê Ê ˆ ˆ N ' , x p + 1 .. x N ) dx p + 1 ... dx N ˜ Y * ' .. x p Ú ( x 1 .. x p , x p + 1 .. x N ) Y ( x 1 Á Á ˜ p Ë Ë ¯ ¯ Ê Ê ˆ ˆ Tr{ G (p) } = N ; Á Á ˜ ˜ p Ë Ë ¯ ¯
QSCP XIII 2008 East Lansing Ê Ê ˆ ˆ w 2 w m - 1 1 . w Á Á ˜ ˜ w 3 w 6 w 3 ( m - 1 ) 1 . Á Á ˜ ˜ i p 1 B = Á Á ˜ ˜ m ; w = e . . . . . m Á Á ˜ ˜ . . . . . Á Á ˜ ˜ Á Á ˜ ˜ 1 w 2m - 1 w 2 ( 2m - 1 ) . w ( m - 1 )( 2m - 1 ) Ë Ë ¯ ¯ h = h 1 ,h 2 ,...h m h B = g = g 1 ,g 2 ,...g m h B - 1 = f = f 1 , f 2 ,...f m
QSCP XIII 2008 East Lansing Coleman’s Extreme State for the Antisymmetrised geminal power g N/2 for N particles, m=2s, where i=1,2..s correspond to a -spin and i+s to b -spin: k = m 1 1 Â g 1 = h k ; h k = f i Ÿ f i + s m 2 k = 1 Y = [ S N / 2 ] - 1 g Ÿ g Ÿ ... = g N / 2 Theorem: The geminal g is an eigenfunction of G (2 ) (g N/2 ) with a nonvanishing eigenvalue if and only if g is of extreme type, e.i. The eigenvalues of G (1 ) (g) are all equal
QSCP XIII 2008 East Lansing The Extreme Case and the emergence of ODLRO Off-Diagonal Long-Range Order m G ( 2 ) = l L g 1 g 1 + l S (2) + G S Â (2) g k g k = G L k = 2 l L = N 2 - ( m - 1 ) l S ; m Æ • ; l L Æ N 2 N ( N - 2 ) l S = ( 2m - 1 ) ; m Æ • l S Æ 0 2m m ≥ N { { 2 ; E = Tr H 1 G ( 1 ) } + Tr H 12 G ( 2 ) } If m = N 2 ; l L = l S = 1. (independent particles)
QSCP XIII 2008 East Lansing m m m h k ( d kl - 1 Â Â Â ( 2 ) µ G S m ) h l g k g k = k = 1 l = 1 k = 2 m m 2 ( e k + e l ) ( d kl - 1 ( 2 ) µ h k e i b 1 e - b L B G S Â Â m ) h l k = 1 l = 1 m - 1 i p m ( k + l - 2 ) ( 2 ) µ ; e - b L B G S Â remember Q kl = ( d kl - 1 m ) e f k f k + 1 k = 1 h if be l = 2 p l 1 m ; l = 1 , 2 .... m ; ( where b = kT and e l = ) 2 t l Requesting the precise relations between the temperature, b , the life time and the dimension of the Jordan block the thermalized matrix becomes proportional to the complex symmetric matrix Q similar to J
QSCP XIII 2008 East Lansing Consequences P t - i ˆ t ; ˆ P = ( w 0 t - i ) ˆ I + ˆ J Propagator : e m - 1 J = h Q h = h B - 1 QB h = f J f = ˆ Â f k f k + 1 k = 1 ˆ J ˆ T = t generates a polynomial evolution k m - 1 J t - i ˆ Ê Ê ˆ ˆ it 1 Â ˆ t = J k e Á Á ˜ ˜ k ! Ë Ë t ¯ ¯ k = 0
QSCP XIII 2008 East Lansing Propagator ‹ Fourier transform fi fi Resolvent P t exp( - i ˆ ) ´ ( wt ˆ I - ˆ ) -1 P t - k m ) -1 = ( wt ˆ I - ˆ Â ˆ J k - 1 P ( ) ( w - w 0 ) t + i k = 0 Power factor ‹ Fourier transform fi fi Higher order poles
QSCP XIII 2008 East Lansing Exponential decay rule - t dN = - 1 t N ( t ) dt ; N ( t ) = e t General rule - t dN = t m - 2 ( m - 1 - 1 t . ) N ( t ) dt ; N ( t ) = t m - 1 e t Note that dN(t)>0 if t<(m-1) t
QSCP XIII 2008 East Lansing
QSCP XIII 2008 East Lansing
QSCP XIII 2008 East Lansing QUANTUM TECHNOLOGY Application Area Mechanism/Technique Coherent Dissipative Quantum-Thermal Systems Correlations Aqueous Solutions Grotthus Type Proton Transfer Self-Dissociation Polar Molecules FIR Molten Salts Ionic Conductance Protons and Muons in Metals Coherent Tunneling Polymers, Organic Molecules Quantum Diffusion
QSCP XIII 2008 East Lansing
CMMSE 2008 Murcia QSCP XIII 2008 East Lansing Applications of the Coherent - Dissipative Ensemble Proton Transfer High TC Superconductivity Microscopic Selforganization E. J. Brändas, in Dynamics During Spectroscopic Transitions, Eds E. Lippert, J. D. Macomber, Springer, Berlin Chapters 6 & 7, (1995). E. J. Brändas, Adv. Chem.Phys. 99, 211 (1997). E. J. Brändas and B. Hessmo, Lecture Notes in Physics, 504, 359 (1998). L. J. Dunne and E. J. Brändas, Int. J. Quant. Chem. 99, 798-804 (2004). E. J. Brändas, Adv. Quant. Chem. 47, 93-106 (2004); 54, 115-132 (2008).
QSCP XIII 2008 East Lansing d l Quote in ”Einstein Defiant”, by Edmund i Blair Bolles, 2004: b r ö Einstein granted that the (Dirac) equation was ”the most s logically perfect presentation” of quantum mechanics yet found, but not that it got us any closer to the ”secret of the t Old One”. It neither described the real world phenomena that a he wanted to understand nor proposed new concepts that l would make the real world accessible to understanding. P
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