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Quantum and Classical Fluctuations Seminar@Riken 2013 July 9 Physics of quantum measurement and its applications I Masahiro Morikawa(Ocha Univ.) Collaboration: N. Mori (Ocha Univ.), T. Mouri (Meteorological Research Institute), A. Nakamichi (Koyama Observatory Kyoto) Quantum mechanics is an excellent theory to describe laboratory experiments. However its operational logic prevents us from applying it to the autonomous systems in which no explicit observer exists. The first-structure problem, i.e. generation of the primordial density fluctuations in the early Universe, belongs to such category. In the first part of the presentation, we propose a physical description of the quantum measurement process based on the collective interaction of many degrees of freedom in the detector. This model describes a variety of quantum measurement processes including the quantum Zeno effect. In some cases this model yields the Born probability rule. By the way, the classical version of this model turns out to be a good model to describe the geomagnetic polarity flipping history over 160 Million years. Then, in the next part, we generalize this model to quantum field theory based on the effective action method of Keldysh type, in which the imaginary part describes the rich variety of classical statistical properties. Applying this method, we describe (a) the EPR measurement as an autonomous process and (b) the primordial fluctuations from the highly squeezed state during inflation in the early Universe, in the same footing. 1 §1 Introduction - Expansion of the Universe New born Universe produces all the structures while it cools: time Matter density vs. Length scale: de Vaucouleurs-Ikeuchi Diagram 10100 Plank time 19 density 90 3 10 10 GeV gr/cm 2 1080 Be c yo r nd 2 (General Relativistic 1070 th 2GL B e e H World) 60 y or 10 o iz G n o d n ( 1050 Q C au u s a al 1040 n li t m u it m ) 1030 U n c 1020 e Quark, nucleus r r t Strong Force ★ 4 a Neutron stars 10 cL i 10 n ★ WhiteDwarf t y Gravity l planet 100 i m atom, molecule, Life ★ Stars i -33 10-10 t Electromagnetic force 10 eV galaxies The Universe 10-20 -2 CBR (Q.M. World) 10 eV clusters 10-30 10-35 10-25 10-15 10-5 105 1015 1025 scale L （cm） Laboratory Physics is connected to the cosmic Physics: 1. Quantum measurement physics – Geomagnetism 2. EPR measurement – Origin of the primordial fluctuations §2 Quantum mechanics in Universe Quantum mechanics is an excellent theory to describe Universe. 22 2 æö pke0 GMç M÷ ■ErpEatom=-,, DD»h =- atom ç ÷ 2mr Rèøç mp ÷ yields 2 h 7 Rplanet =»´1.0 10 m eGkmm0 ep ek33/2 0 26 →Planet（Jupiter） Mplanet =»´3/22 8.1 10 Kg Gmp ■ go beyond the p-p coulomb barrier by quantum uncertainty: (coulomb energy)=(energy uncertain) i.e. 22 p h 4pGM 2 Enf ==2 and gravitational energy Egr = , 2mp 2xmp x which yields T =´510K8 . Furthermore, 2 h 8 Rstar =»´3/2 1/2 3.2 10 m eGkm0 ep m 33/2 →star(sun) ek0 31 Mstar =»´3/2 3/2 1/2 2.3 10 Kg Gmep m ■ relativistic limit vm®®c, epm yields the white dwarf and the neutron star, the structure of degenerate fermion 3/2 3/2 c h 30 M =»´3/22 1.3 10 Kg 22Gmp 3/2 h 3 R =»2 10 m 22cGmp →neutron star - Classical limit h ® 0 is not realistic at all. - Macroscopic structure is based on the microscopic quantum theory. - The specific scale appears reflecting the non-extensivity of gravity. 3/2 h 25 - The structure by degenerate neutrino : R =»´2 3 10 m 22cGmn (almost cosmic radius) - Case of Boson: if CDM is the boson of m £ 20eV, then the boson is degenerate aG& 2 8p H 2 ==r ++r r ( ) 2 ( g f l ) ◆ This BEC field behaves a 3c as classical fields. rrr&ggg=-3H -G ◆ If the potential V is rr&ff=-6H ( -V ) +GGrg - ¢ r f unstable, the BEC suddenly r&l = -3H rl + G¢rf collapses. ◆ The force V ¢ and the condensation force G balances w.e.o.⇒constant Lnow（acc. Exp.） PTP115 (2006) 1047, Phys.Rev.D80:063520,2009 0.00004 -V ¢ Grg 0.00002 L V 0 now 0.00002 0 0.020.040.060.080.10.120.14 §3 Quantum mechanics in the universe - Deeper connection - Quantum mechanics is an excellent theory to describe Universe. - But in more fundamental level, autonomous irreversible Universe conflicts with the hybrid structure of QM, i.e. Schrö dinger equation + measurement process - The ultimate description of the Universe is thought to be the wave function of the Universe Y (gmn,,,...jy ) , which obeys Haˆ Y=( ,0j ) （mini-superspace app.） (Wheeler-DeWitt eq. ←to operator←H = 0←canonical gravity) - How to set initial (boundary) condition? - No observer can reduce the wave function! - The Universe never repeats. No one can reset the Universe for repeated measurements. - A star begun to shine when someone once measured the star ?? - Primordial density fluctuations in the early Universe. k-mode of inflaton obeys ¢¢ 222Ht , and the vkHevkk+-( 20) = --1/2 1 Ht squeezing amplifies the fluctuation: vkk »+(1 ikHe) 2 ffˆˆ(xy) ( ) = dj( k) ®® Pk( ) 144444424444443k 144424443 quant um mechani cal cl assi cal A k-mode becomes classical when it goes beyond the horizon?? Does quantum fluctuations make spatial structures?? Are qu. fluctuations equivalent to the stat. fluctuations?? ◆ Origin of the problem is the hybrid structure of QM: (ab­+ ¯) AAA Þ a ­ + b ¯ 0 1444444442444444443­¯ Yentangled ì ï ¾¾¾®­2 A­ ï a ï ï or Þ íï measurement ï ¾¾¾®¯2 A¯ ï b ï ï either îï Schrö dinger equation + measurement process system & apparatus observer POVM This hybrid structure is an axiom of QM: A wave function shrinks into an eigenstate when measured. §4 Many world -a typical unified theory of QM - cosmologists' favorite ◆ One-to-many map (Observer's indeterminism) can be brought into an one-to-one map if the domain is extended. i.e. state reduction is regarded as the bifurcation to many correlated pairs (of a system and the observer) world1 a ­ A­ (ab­+ ¯) AAA0 ® a ­­¯ + b ¯ world 2 b ¯ A¯ Evallet Wheeler Hartle Gelmann… ◆ Problems of many world theory 1 bifurcation is ambiguous 11 Y=­¯-¯­=¬®-®¬( ) ( ) entangled 22 A special base is needed（Zurek） 2 bifurcation dynamics is missing: When/How bifurcate? 3 How the decoherence between the worlds realize? 4 No intrinsic prediction of the theory; We cannot demonstrate the theory. ◆ A bit qualitative argument: Consistent historytheory (decoherence between the worlds) Dqqqq(aa, ¢¢¢) =- d dd( ff) òòaa¢ ´-exp(iSq{ [ ¢¢(ttr)] Sq[ ( )]} /h ) ( qq00 , ) coarse grain Q of the full set of variables qxQ= ( , ) to yield the effective dynamics for x. If D (aa,0,¢¢) =¹ a a, then two histories aa, ¢ decohere and the consystent probability is assigned to each history. M Gell-Mann, jB Hartle - Physical Review D, 1993 - APS Actually, this is the Feynman-Vernon Influence Functional Theories or Caldeira-Legett formalism in our world. R. P. Feynman and F. L. Vernon, Ann. Phys. (N. Y.) 24, 118 (1963) A. Caldeira and A. J. Leggett, Phys. Rev. Lett., vol. 46, p. 211, 1981 Therefore many worlds may be in our own world. §5 Simple model of quantum measurement - particles on a ring (1-dim.) measurement apparatus determines the location of a system particle: either right or left side of the ring. - a system measured: yq0() a prticle wave function yqi-1() V0 (q ): ptential yqi() yqi+1() L R - measurement apparatus: many particles yqi(),iN= 1,2,...., - Equation of motion: iN= 0,1,2,...., where : common potential : attractive force : order variable i.e. meter readout - initial condition for the apparatus yqi( ),iN= 1,2,...., where xxii, ¢ are Gaussian random variables( dispersion s) i.e. all the particles are set almost on the potential top q » 0. - all the particles yqi() obey linear equation of motion - dynamics of apparatus yqi(),iN= 1,2,...., m ==1,h 0.02,sl = 0.01, =- 1 ⇒ attractive force yields an almost single wave packet - dynamics of apparatus yqi(),iN= 1,2,...., l = 0 ⇒ the wave packet dissipates away if no attractive force The condition for the wave packet to form a single peak: i.e. attractive force dominates kinetic energy/differential force - case of system (located right)+ apparatus (a = 0) - case of system (located left)+ apparatus (ap= /2) - case of system (located both sides)+ apparatus (ap= /4) only a single case happens The condition for the proper measurement : collective mode never tunnel ＆ a single wave function tunnels i.e. NmEamV?h//2//2 V( ) » h ¢(s ) ⇒ Coexistence of the extremely separated time scales will be necessary for proper measurement. - case of system (general superposition)+apparatus(0/2££ap) only a single case happens - Probability distributions for 0/2££ap frequency（after measurement） --- frequency ratio of meter readout is `right` --- error ratio (inconsistent meter readout and the system state) … Born probability rule 2/ap(system state bef. measurement) - Born rule appears for this special model, but no guarantee that the relation becomes straight for N→∞ and error→０ ★ A condition for faithful measurement The balance between 2 (1) jq(t, ) can freely move on V0 (q )(randomness) and ( ) 2 V (q ) (2) systemyq0 t, can trigger jq(t, ) through HF (trigger) i.e. æö2 2 ssVN¢( ) » lç e-5( 1/2) ÷ 0 èøç ÷ ■ Both classical and quantum nature of apparatus are simultaneously needed for the quantum measurement. §6 Classical analog: geomagnetic flipping - The same model describes the geomagnetic flipping history ↓present S-N 160M years ago↓ http://www.n-tv.de/wissen/Zeigt-der-Kompass-bald-nach-Sueden-article935939.html - Classical version of the Quantum measurement model describes the dynamics of geomagnetic variation including flipping ■ Long-range Coupled-Spin model (LCS) NN N r N N 11r& 22& r 2 r r Ks==ååiiq , V =W×+ml å( s i) ååsij× s 22ii==11= i 1 ji=11= dL¶¶/ q& ( i ) ¶L & LKV=-, = -+kqi x dt ¶qi 1 r æöN r MsºW×ç ÷ mean field: çå i ÷ N èøç i=1 ÷ This model describes random polarity flip despite being a deterministic model.
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