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. C

Computer simulations of the basic MHD equation yields vortex dynamo structures → LCS model Kida, S. & Kitauchi, H., Prog.r Theor. Phys. Suppl.130, 121 Ñ=v 0 r æö¶v rr1 r r 2 r rrrnç +×Ñ=-Ñ-vv÷ p W´ r + D v A A 000èøç ¶t ÷ ( 2 ) C r rrr r A C +-rrgvjB2 0 W´+´ r C A Ñ×B = 0 A r C ¶B r rr C =Ñ´vB ´ +h D B ¶t ( ) ¶T r +×Ñ=D+vTke T ¶t rr1 JB=Ñ´ m0 - Geomagnetic polarity flipping history over 160My: ↓now S-N superchron 16oMya↓

- LCS model calculation:

almost steady period(: 105y) and rapid polarity flipping(: 103y) coexists Power spectrum Interval distributions

observation

LCS model A typical polarity flip dynamics: - Physical backgrounds: HMF model (=Hamiltonian Mean Field) deterministic model which shows phase transition

T N r 2 må(W×si ) ® 0 i=1

- yields phase transition e - Core-halo structure is formed Core: almost stable dipole ↔meter jq(t, )2 Halo: rapid variation ↔system yq0 (t, )

spots of reverse polarity moves rapidly

magnetic field distribution on the core-mantle boundary (upper: 1980, lower: 2000) - Kuramoto model analogy (describe synchronization)

l ek/ » T mW2

& N ¶qi K Kuramoto model =+wqqijiå sin( - ), 1 ££iN ¶tNj=1 synchronization: An interaction between many oscillators yields a single stable rhythm globally. §7 Other planets, satellites, and stars - balance of generation/diffusion of magnetic fields, balance of collioris force and magnetic pressure, yield dipole moment:

- plot of d for various planets, satellites, and the sun

yields the scaling:

This scaling implies for the sun, #Taylor cell: §8 The sun solar activity ~ magnetism ~ #sun spots, polarity flips every 11 years

almost 11 years

(fake: month spin period)

1/f fluctuation + 11y period - LCS model with N=101

clear period and f -1.2 fluctuation synchronization of element spins? http://xxx.yukawa.kyoto-u.ac.jp/abs/1104.5093 A. Nakamichi, H. Mouri, D. Schmitt, A. Ferriz-Mas, J. Wicht, M. Morikawa

■ The same model describes the dynamics of celestial magnetic fields including polarity flip. §9 summary 1． - No observer exists in the Universe and QM should be described without external operation. - We proposed a model of quantum observation, as a physical process . - coexistence of extremely separated time scales. mean field and the system(attractive force/many particles) - localized mean field describes the classical meter - process of measurement: positive field back 1. the system triggers the mean field 2. the mean field synchronizes with the system - Born rule is derived in restricted cases % 2 - Ozawa quartet {KX,,,a U} is yi Î K , X% = j ,a:initial random distribution,U :Unitary evolution(HF app.) 2． The classical version of the same model describes geomagnetism - coexistence of extremely separated time scales. core and halo (←spin-spin interaction) - almost stable dipole component = localized mean field - rapidly moving component = individual mode - process of polarity flip: positive field back 1. halo triggers the core 2. the core synchronizes with the halo - planets, satellites, stars may have common physics of their activity and magnetism

■ Core-halo structure is common, in QM, celestial magnetism, gravity... Seminar@Riken 2013/July 9 Physics of quantum measurement and its applications II

Masahiro Morikawa(Ocha Univ.)

Collaboration: 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 – Universe and QM, Stat Mech. - cosmological structures are not separated with each other but micro-macro scales are connected. §2 Statistical and Quantum mechanics - Entropy and action ...analogy Skentropy=- BTr(rr ln ) → maximized Siaction =-h ln Y → minimized - Then the entropy for a dynamical system is implied as æö ÷ k ç ÷ SSS»-B Imç *(time reversal ) ÷ entropyçå( action action )÷ h ç 1444444444444442444444444444443 ÷ èøç CPT-G effective action % ÷ k where: S »GB Im% entropy h æö æö% ç ÷ ç G ÷ ç i %%Re1 Im ÷ expçi ÷ =G-G expç ÷ èøç hhh÷ ç { ÷ ç ÷ èøSkentropy/ B G% Im: will imply statistical information (probabilistic) → a system actually evolves toward its maximum G%Re: will imply mechanical information (deterministic) → only the extremum is relevant for evolution

- G¹% Im 0 ← finite temperature/density, particle production, unstable systems, ... - for the case of (meta-)stable vacuum, G=% Im 0 if time scale is not considered ■ relation with the Caldeira-Leggett model (1980) Full evolution interaction: Coarse grain Q and yields effective action for the reduced density matrix for q:

+ -

↔ G% Im: fluctuation

↔ G%Re: dissipation ■ generalized closed contour path integral method(CPT) Keldysh1965 ZJ%%[]ºº Tr[(Exp[ T% i ò J %%fr ])] Exp[ iWJ % []] % ZJ%%[ ]=- Tr[ T% (Exp[ i òò J %%ffr i V [ ]] )] d i =Exp[-iòòòò V[ ]]Exp[- JxG%% ( ) ( xyJy , ) % ( )]Tr[:Exp[ i J %%fr ]: ]. 1dJ% 2 0 % % LL[]fff=-0 [] V [], Jx()=- J+ () x J- () x, ff()xxx=-+ () f- () %% òòJdxJxxdxJxxff=-++() () ò-- () f ()

æöGxyGxy(, )+ (, )÷ ç F ÷ GF Gxy% (, )= ç ÷ 0 çGxyGxy(, ) (, )÷ èøç - F ÷ G- ˆˆDk()--- iBk () Dk () iBk () GkF ()== , Gk () , Dk()22++ Ak ()F Dk () 22 Ak () ˆ Dk()m iBk () G Gk±()=- i . + 22 G Dk()+ Ak () F D(k): renormalozation for the wave function and mass R A(k): odd for time reversal, new in CTP S-ÛF (xx¢) Im a (friction, irreversible) I B(k): even for time reversal, new in CTP S-ÛF (xx¢) Re a (classical fluctuations)

% dW ˆˆéù4 Defining the order variable jff%()xTyidxJxxº=YYii,,tt( ) exp ( ) ( ) ii dJ% ëûêúò and effective action Gº%%%[]jj%%WJ [] -ò J , we have

iG%[]j %Re 1 -1 ed= []x P[]x Exp[i (G+xjD ) ] where P[]x =-Exp[xxB ] ò 2 òò %Re dG % Further =-Jxc () follows and becomes the (classical) Langevin equation: dj%D()x t x 2 ( ) ¢¢¢¢¢( ) ()¶¶x +=-+-+m j x Vòò-¥ dt dx A ()() x x j x x x - In general. P[]x includes non-Gaussian terms - `We do not extremize G% Im (as it is statistical factor)` is a big assumption. - j (x ) and x(x ) are classical fields §3 Physics of quantum measurement

ì ï ¾¾¾®­2 A­ ï a ï (ab­+ ¯) AAA Þ a ­ + b ¯ ï or 0 1444444442444444443­¯Þ íï Y measure ï entangled ¾¾¾®¯2 A¯ ï b ï ï either îï

3-1. one-to-many (non-deterministic time evolution) 3-2. mixed state ⇔ pure state intergradation 3-3. quantum and classical variables coexistance 3-4. from quantum to classical evolution 3-1. one-to-many

- extension of the domain (=many world) makes the map one-to-one although non-deterministic bifurcation should be described

- A typical process would be the dynamics of spontaneous symmetry breaking or phase transition 5

0

-5  

 -10 - This is essentially infinite degrees of V -15 freedom -20

-6 -4 -2 0 2 4 6  - Thus, we need QFT beyond QM → vacuum changes in time 3-2. Mixed state ↔ pure state intergradation - a spin (½) in the environment yields pure ↔ mixed evolution dr =-iSwr[,( t) ] + aS[ r( tS) ,] + bS[ r( tS) , ] dt 3 +- -+ ++cS[ 33r ( t),.. S] hc - If (spin temperature) < (environment temperature) r evolves from pure ⇒ mixed well studied relaxation/decoherence process - If (spin temperature) > (environment temperature) r evolves from mixed ⇒ pure - Actually anb==+,1 n and ab=-exp[ lhEint /(kT )] æöb Thus l >®0,T 0 yields u ç 0 ÷ çab+ ÷ l <®0,T 0 d r ® ç ÷ yields t®¥ç a ÷ ç 0 ÷ This is pro-coherence. èøç ab+ ÷ 3-3. Quantum & classical variables coexistence Partition function of QFT connects classical and quantum:

Jx( ) ZJ[ ] Jx( ) Jx( ) ZJ[ ] =Y,exp t Téù i Jfˆ Y , t ffëûêúò ii K Jx( )

j (x )

G[j ] ( ) Also the effective action G[j ] j (x ) j x

K j (x )

… These are the whole collection of fluctuations: j (xd) YY,,txAfˆˆ( )K ˆ(yz)f( ) tj (ｚ) xdydz ò { 1444444444444442444444444444443ff ii{ classicalquantum classical 3-4. Emergence of classical variable in QFT - Keldysh effective action yields stochastic eq. for j (x ): 1 - ImG=% (even in jjj (xxBxyy )) = ( ) ( - ) ( ) + ... DDD2 òò This kernel Bx()- y is positive. Therefore within Gaussian approximation, the effective action can be written as 1 edPiiiG[]j =G+[][]Exp[Rexx xj ],PB[]xxxº- Exp[-1 ] ò D 2 òò is the Gaussian weight function for random field x()x . On the other hand, Re G+xjD ºSeff is real, and the variational principle for jD()x is applied to yield % ddjSxJeff /()D =- which is the Langevin eq. for classical fields: 2 ¢¢¢¢¢t ()W+=-+mVdtdxAxxxjjxccòò-¥ ()() - + 3-5. Common origin for all the above problems?

Order variable is 1. SSB defined and evolves 4. Emergence of classical variable

Phase transition Classification of proceeds classical and quantum

Physical description for the 2. qu⇔cl quantum 3. coexistence of qu/cl measurement →irreversibility §4 Quantum measurement model in QFT - starting points - measurement apparatus is the fields - all dynamical process, including measurement process, is caused by the field interactions. - information obtained by a measurement is a pattern on the field

- A simple model of measurement 5 0

-5  

 -10 r V Field f measures a spin : -15 S -20

2 -6 -4 -2 0 2 4 6 1 2 m l rr  L =¶( f) -ff24 - +mfS × B +(bath) where m > 0. 224!m Progress of Theoretical Physics Vol. 116 No. 4 (2006) pp. 679-698 : Quantum Measurement Driven by Spontaneous Symmetry Breaking Masahiro Morikawa and Akika Nakamichi - The minimu model which keeps the essence of the original mdel.

- Generalized effective action method yields for fjdfˆˆ=+ , l rr jgjj& =-3 + mSB ×+ x, xx(tt) ( ¢¢) =- ed( tt) 3! - spin density matrix in the environment, N. Hashitsume, F. Shibata and M. Shingu, J. Stat. Phys. 17 (1977), 155. dr =-iSwr[,( t) ] + aS[ r( tS) ,] + bS[ r( tS) , ] dt 3 +- -+ ++cS[ 33r ( t),.. S] hc rr ■ x triggers SSB initially SB×>0 ⇒ linear bias ⇒ force j toward

j+ > 0 3 2 

 1 

V 0 1 2 4 2 0 2 4 j-  j+ ■ mixed→pure in the factor ab=-exp[ hmj(tB) /( kT)] , the effective temperature Tt/j ( )→0 ⇒ spin is brought into a pure state↑ ­

r ■ positive feedback linear bias is enhanced furthr ⇒ spin tends to be r S more parallel to B . - This positive feedback fixes the spin ↑, order variable jj® +. rr ■ On the other hand if initially SB×<0, then the above process fixes the spin↓, order variable j-. ■ results: we solved the coupled equations for r and j. æö10 ç ÷ If r (0) = ç ÷, then èøç00÷ æö00 ç ÷ If r (0) = ç ÷, then èøç01÷ Further, æö0.4 0.49 ç ÷ If r (0) = ç ÷ etc. èøç0.49 0.6 ÷ then, the frequency distribution yields QM results jr(tt) ( ) Tr r( t)2 - comments 1.`world bifurcation`(many world)→`SSB`(our world) rr 2. For the general initial condition ab­+ ¯, bias dmg=×( / ) SB affects Pt(j, ) non-linearly. a faithful measurement needs a calibration. 3. a relevant time scale: -1 1 éæg e 2 öù t0 =+ln êúç d ÷ 2gggëûêúèøç ÷

i.e. the completion time needed for SSB.

4. Application to various measurement processes are possible. rr §5 EPR measurement (two spins SS12, )

5 5 2 0 0 -5 -5

2 

1 m l   

 -10

 -10 24 V V L =¶fff - - -15 -15 ( m ) -20 jj2 ® - -20 jj1 ® +

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 224!   rr t ++mfS × B (bath) j 2 B2 B1 j1 - spatially separated spins r r - localized apparatus ff12, measure S2 S1 each spin under individual xvt1 = xvt2 =- magnetic fields BB12, . are applied - only causal interaction x - initial state of the system measured: 1 y =­Ä¯-¯Ä­( ) 2 1212 - Correlation terms in generalized effective action rr rr ò mf(xSx) ( ) × Bxdx( ) →ò mjTr( rSBxdx× ( )) ))))rr rr )) mfSB×× mf SB 2 òò ( ) ( ) →ixBxSxSxBxxòò mj( 11) iijj( )Tr( r( 12) ( )) ( 22) j( ) x1 x2 ...yields stochastic diff. Eequations: l rr rr jgjjmr& =-3 +Tr SB ×+× x B 113! 1( 1111) l rr rr jgjjmr& =-3 +Tr SB ×+× x B 223! 2( 2222) ˆˆ(12) ( ) xx12ij ← Tr(r Sij S ) (-dij intially, quantum corelation)

For the state rrºÄ12 r dr =-iSwr[,( t) ] + aS[ r( tS) ,] + bS[ r( tS) , ] dt 3 +- -+ ++cS[ 33r ( t),.. S] hc 1 - From the initial state y =­Ä¯-¯Ä­( ), 2 1212 rr {BB12== (0,0,1), (0,0,1)} or rr {BB12== (1,0,0), (1,0,0)} etc. complete anti-correlation is obtained.

Spin state r and meter read j are consistent with each other. ■ Bell in-equallity is violated?

- in probability PB(,12 B )= ò jljlrl 1final( , B 1) 2 final ( , B 2) d the weight r depends on jj1122,,,BB. Therefore this is not the `hidden variable theory`.

ˆˆ(12) ( ) - xx12ij ← Tr(r Sij S ) all the correlations of quantum theory is included.

- Evolutions of the classical field j and the density matrix r couple with each other. Positive feedback is the essence. - Actually rr -1/2 dx=×11Bt/, g 2 e( ) » g (g :friction M. Suzuki, Adv. Chem. Phys. 46 (1981), 195.) and rr (12) ( ) PBPB+-( 12) ( ) = P +- yield rr CBB( 12, ) º+-+ P++ P --( P +- P -+ ) rr rr --1/2 1/2 =×erf(gx11BB) erf ( gx 22 ×) . rr »-g-1BB × 12rr rr Setting the magnetic field BB12= g makes CBB( 12,) »- cos q 12. Thus if the `accuracy` (erf( ) ,g ,...) are sufficient, we have rr rr rr rr CBB( 12,,,,2) ++- CBB( 12¢¢¢¢) CBB( 12) CBB( 12) >

■ There is a possibility that the Bell inequality is violated. §6 Origin of the primordial density fluctuations ■ The inflationary model in the early Universe スケール

- The scenario: from FRW expansion（at( ) µ t） then, de Sitter expansion（at( ) µ eHt ） connected to FRW（at( ) µ t）

t t_i t_f t_now - Inflaton field (scalar field) promotes the inflation

¢¢ 222Ht - k-mode obeys vkHk +-( 20e )vk = and highly squeezes, yielding IR-fluctuations. --1/2 1 Ht vkk »+(1 ikHe ) （vakkº j ） ■ Universal inflation model? V (f) V (f ) V (f) A standard model but allows negative region of the potential. Then, this model f f f always enters into the period of stagflation (rrff===& H 0 but j ®¥), where the expansion stops. Then the uniform mode j0becomes unstable and decays → FRuniverse ⇒L-self regulation Phys.Rev.D80:063520,2009 2 ffˆˆ(xy) ( ) = dj( k) k 144424443 vk ■ standard theory assumes 144444424444443 , jk º quantum theory clasical a stat.. mech i.e. a mode becomes classical when goes beyond the horizon. 3 2 2 4pkvk H Therefore Pdj =¾¾¾¾¾®(scale invariant) (2p)3 a k/0( aH )® (2p )

each mode evolves analogously → scale invariant fluctuations 1/H

k_1 k_2

l_0 t_1 t_2

t ■ quantum → classical fluctuation: analogous to EPR measurement

de Sitterexpansion

horizon size (causal limit) squeezed state (after complete measurement)

If at x1, then at x2 or,

If at x1, then at x2 ■ Interaction in the Universe at reheating stage.

5 rr 0 -5 

mjSB 

 -10 V -15

-20

-6 -4 -2 0 2 4 6  case of spin 2 lfkk j j0

4 interaction lf , fj=+0 dj 2 2 the term jdj0 (k ) jdj0 (k ) yields

%% 22%% G=G00 +ò ljdj%%(k ) f(xy)f( ) ljj %% 0( -kdk) r ¥ sin[kz ] éæhh¢33- ö ù Reff(z ) ( 0) =+dkêúç ÷ O k 3 ò 2 ç ÷ ( ) 0 4p z ëûêúèøç 3hh¢ ÷ ...IR(k ® 0) converges →Gret: mechanics

éù-11æöhh¢ êú-+ç ÷ r ¥ sin[kz ] 2 ç ÷ Im ff(z ) (0) = dk êúhh¢ k 2 èøh¢ h ò0 4p2z êú êú+Ok2 ëû( ) … IR-divergent →Gc: statistical mech.

yields scale free fluctuations ■ Applying the least action method onRe G% , (not Im G% , stat. mech. )

tx Wjj(x) =- V¢( ( x)) + dyGret ( x - y) jx( y) + ( x ) ò-¥ 2 2 2 where xx(xy) ( ) =- Gxyc00( ) ljj( x) ( y) IR-divergent statistical fluctuations

Leaving the most relevant part jx&&kk» , we have 3 2 4pkH2 2 4 Ptdj =¾¾¾¾¾®Djljk ( 0 ) (2p)3 k/1( aH )= (2p ) i.e. scale invariance is universal but the amplitude is not H 2 present case, setting D»tVjj00/,& ( j 0) » 0 yieldPO= (1) dj (2p ) 4 chaotic lf model (j0 » 0) yields no fluctuations In general, the amplitude depends on the interaction at reheating ■ quantum theory in the inflationary era

- correlation in the quantum system fˆ(x ) is traced in the macroscopic r pattern in apparatus j (tx, ). This evolves into the present large scale structure. i.e. the quantum system fˆ(x ) disappears and only apparatus remains in the universe.

- IR-divergence in GGRe, Im : does it factorize? ...not solved éùd éù1 ZJ[ ] =+expêú i L exp êú JG( Re G Im )J ëûòò(iJd ) ëû2

- variety of reheating mechanism → selection of the inflation models §7 summary

1． We constructed a physical quantum measurement model. - phase transition (SSB), de/pro coherence, positive feedback - measurement process is a phase transition (SSB) r it is deterministic (-V ¢) as well as non-determinsitic(x(tx, )) - EPR measurement process, Bell inequality 2． Origin of the primordial density fluctuations - classical apparatus variable determines the large scale structure - statistical fluctuations are autonomously generated as in the case of EPR measurement. - density fluctuations depends on the reheating interaction and the potential ■ Cosmological structures are not separated with each other but micro- macro hierarchy are connected: measurement process, EPR, celestial magnetism, primordial density fluctuations ...

- A system can be non-equilibrium and dissipative at any scale.