seminario geometry-qm-2

The Geometry of Quantization (II)

Phys. Rev.Lett. 105, 150606 (2010)E.G. and C.Pagani

Ennio GozziDept. of Physics

Theoretical SectionUniversity of TriesteStrada Costiera 11Miramare-Trieste

MOTIVATIONS

● Superposition of states is one of the main feature of Quantum Mechanics (QM). It is at the heart of typical quantum mechanics effects like interpherence, entanglement etc.

● In Classical Mechanics (CM) instead we do not have superpositions and this creates all the troubles we encounter when we try to couple a quantum mechanical system with a classical one.

● In this work we shall try to understand what kills the superposition in Classical Mechanics.This will shed a new light not only on CM but also on QM.

● Path integral counterpart of the operatorial formalism mentioned above (CPI), which is different from the quantum one (QPI);

● Geometrical analysis of the space where the path integral formulation lives. Introduction of two Grassmann partners and to time t and of supervariables;

! !

-1-

Contents

● Koopman and von Neumann Hilbert Space formulation of classical mechanics (KvN) 1931;

● Local invariances;

● Observables;

Superposition principle in KvN?;●

● Solution of the problem.

Classical Mechanics

(Review of Hamiltonian formalism)

● Hamiltonian: H(!)

● Poisson brackets: {A1(!), A2(!)}pb = "aA1#ab"bA2

● Probability density distribution in phase space:

microcanonical canonical

!("a, t) ! #[H " E]

!("a, t) ! exp["#H]

● Phase space: !a ! (q1 · · · qn; p1 · · · pn), a = 1, · · · , 2n

● Evolution of observables: dO

dt=

!O

!t+ {O,H}pb

● Equations of motion: !a = "ab#bH

● Evolution of probability density:d!

dt= 0 =! "!

"t= "{!,H}pb = !!a"#ab!bH

-2-

Classical Mechanicsin Operatorial Form

Koopman, von Neumann (1931)

● !("a,!t) = !("a ! #ab$bH!t; 0)

Same functional form but different arguments.

!

● Propagator:

!(", t) =!

d"i K("; t|"i; ti)!("i, ti)

K(!f ; tf |!i; ti) = "[!f ! #cl(tf ;!i, ti)]

is a solution of Hamilton equations of motion.

!cl(t;"i, ti)

-3-

●(Liouvillian)L = i!aH"ab!b = i!qH!p ! i!pH!q

!"

!t= !{", H} = !aH#ab!b" " !iL"

Koopman-von Neumann(KvN) Postulates

● As L is linear in the derivatives one can derive:

i!"

!t= L"

-4-

|!(")|2 = #(")

Probability density in the phase space.

3):

The evolution of ! is via the Liouvi llian L

i!"

!t= L" L = i!aH"ab!b = i!qH!p ! i!pH!q

4):

KvN introduced a Hilbert space made up of complex and square integrable functions on the phase space and imposed the following two postulates:

!(") ! L2

1)

2): [q, p] = 0

The observables? 5):

Classical Dynamics

!Same equation !

i!

!t"(#, t) = L"(#, t)

i!

!t"(#, t) = L"(#, t)

is linear in the derivatives. L

Quantum Dynamics

i!

!t"(x, t) = H"(x, t)

Classical Dynamics

same equation!

!"#

"$

i!

!t"(q, p, t) = %H"(q, p, t)

i!

!t#(q, p, t) = %H#(q, p, t)

Why? %H is first order in !q and !p:

%H = !i!pH(q, p)!q + i!qH(q, p)!p

Quantum Dynamics

di!erent

equations

!"#

"$

i!

!t"(x, t) = %H"(x, t)

i!

!t#(x, t) = ! !

2mdiv("!$"" ! "$""!)

%H = ! !2

2m

!2

!x2+ V (x) is second order in !x

3

!Different Equations

H = ! h2

2m

!2

!x2+ V (x)Reason: is second order in !x

-5-

Why?

Classical transition amplitude via path integral

As the evolution of the and the is the same, the propagator will be the same:

! !

K(!f ; tf |!i; ti) = "[!f ! #cl(tf ,!i)]

So in the continuum limit we get

K(!f ; tf |!i; ti) =! !f

!i

D!!!D"DcDc exp i

!dt "L

K(!f ; tf |!i; ti)

= limN!"

!"

#

N!1$

j=1

%d!j"[!j ! #cl(tj ,!i)]

&'

(!["f ! #cl(tf ,"i)]

!["j ! #cl(tj ,"i)] = !["a ! #ab$bH]tj

det[!ab "t ! "b(#ac"cH)]tj

!d!a ei!a["a!#ab$bH]

!dcadca e!ca[!a

b "t!"b(#ac"cH)]cb

!L = !a["a ! #ab$bH] + ica[%ab $t ! $b(#ac$cH)]cb

-6-

Path Integral and Operatorial Formalism

●

Path Integral Classico

!a ! (q, p) variabili di spazio delle fasi

"acl(t;!i) cammino classico.

Risolve !a = #ab$bH(!)

Z ! "!f ; tf |!i; ti# =!D!!! %(!a$"a

cl(t;!i))

Proprieta delle delta di Dirac:

!("a ! #acl(t; "i)) = !("a ! $ab%bH) det(%t!

ac ! $ab%b%cH)

Esponenziazione:

Z =!D!!!D&DcDc exp i

! tf

ti

dt "L

"L = &a!a + icaca $H

H = &a#ab$bH + ica#ab$b$dHcd

4





Z ! "!f ; tf |!i; ti# =!D!!! %(!a$"a

cl(t;!i))



ac ! $ab%b%cH)

Esponenziazione:


! tf

ti

dt "L



4

where:

How can we turn into an operator?





Z ! "!f ; tf |!i; ti# =!D!!! %(!a$"a

cl(t;!i))



ac ! $ab%b%cH)

Esponenziazione:


! tf

ti

dt "L



4

● Commutators:![!a(t),"b(t)]" # lim

!!0!!a(t + #)"b(t) $ "b(t + #)!(t)"

With our the only non-zero!L = !a"a + icaca + · · ·commutators are:

![!a(t),"b(t)]" = i#ab , ![ca(t), cb(t)]" = #a

b

classical mechanics.![!a(t),!b(t)]" = 0In particular which confirms

that we are doing ●

-7-

!a !" !i"

"#aca !" !

!ca

A particular realization of the commutator above is:

and as multiplicative operators. !a, ca

●

Path Integrals --> Operators

!a(t) !" !i"

"#a

(Liouville operator)H = !a"ab#bH !" H = !i"ab#bH#a = L

!

Path integral Operator formulation (Koopman-von ! Neumann 1931)

! !

! !f

!i

D!!!D" exp i

!dt "L !" exp!iHt = exp!iLt

Question: If the bosonic part of is related to the Liouville operator, what is the meaning

of the full ?





Z ! "!f ; tf |!i; ti# =!D!!! %(!a$"a

cl(t;!i))



ac ! $ab%b%cH)

Esponenziazione:


! tf

ti

dt "L



4





Z ! "!f ; tf |!i; ti# =!D!!! %(!a$"a

cl(t;!i))



ac ! $ab%b%cH)

Esponenziazione:


! tf

ti

dt "L



4


-8-

c- variables and Jacobi fields


generates the evolution of both the points of the phase space and their first variations.

Campi di Jacobi

!"a(t)

•

!"a(0)

•

!""#

""$

d

dt(!"a) = #ab$b$dH(!"d)! equazione di !"a

d

dtca = #ab$b$dHcd ! equazione di ca

ca " !"a

• H genera l’evoluzione sia dei punti dello spaziodelle fasi che delle loro prime variazioni.

• H: le sue proprieta spettrali forniscono informazionisulle poprieta dinamiche del sistema (ergodicita,caos, esponenti di Lyapunov).

14

ca(0) ca(t)

[Fig. 1]

1

!c(t)c(0)" # exp!t

Lyapunov exponents.

-9-

Equations of motion:

standard equations of ! motion for

!!

● Variation with respect to ! gives !a ! "ab#bH = 0

Variation with respect to ● c gives ca ! !ac"c"bHcb = 0

have the same evolution equation as the first variations of ca !a

Jacobi fields.

(!"a)! #ab$c$bH(!"c) = 0

c and differential forms

●

Forme Differenziali

Evoluzione temporale infinitesima!"#

"$

!a! = !a + "#ab$bH

ca! = ca + "#ab$b$dHcd =$!a!

$!bcb

ca si trasforma come una base per le forme ! ca " d!a

%(!, d!) = %0(!) + %a(!)d!a + %abd!a # d!b + . . .

$%(!, c) = %0(!) + %a(!)ca + %abcacb + . . .

%(!, c) e una forma di!erenziale

H%(!, c) = Lh%(!, d!)

Lh: derivata di Lie lungo il flusso hamiltoniano

(Cartan 1900).

13

Infinitesimal time evolution:

ca transform as a basis for forms;

T !!Mca = d!a ! basis of

ca · cb ! d!a " d!b

!

●forms on M

F (p) =1p!

Fa1...apd!a1 ! · · · ! d!ap "# F $ 1p!

Fa1...apca1 · · · cap

!" functions of and ! c

November 4, 2003 17:17 WSPC/139-IJMPA 01598

Cartan Calculus via Pauli Matrices 5233

where the Lagrangian is given by

L = !a"a + icaca !H , with H = !a#ab$bH + ica#ac$c$bHcb . (2.5)

From the Lagrangian (2.5) one can derive, besides the standard Hamilton’s equa-tions (2.1) for ", the following equations of motion for the Grassmann variables:

cb = #bc$c$aHca ,

˙cb = !ca#ac$c$bH .(2.6)

So infinitesimal transformations generated by H are given by

"a!= "a + %#ab$bH ,

ca!= ca + %#ac$c$bHcb =

$"a!

$"bcb ,

c!a = ca ! %cb#bc$c$aH =$"b

$"a! cb .

(2.7)

We notice from these equations that ca transforms, under the di!eomorphismgenerated by H, as a basis for the di!erential forms d"a, while ca transforms asa basis for the vector fields !

!"a , see Refs. 1 and 7. From the kinetic part of theLagrangian (2.5) we can derive, as Feynman did for quantum mechanics, the gradedcommutators of the theory. They are:

"["a, !b]"# = i&ab , "[cb, c

a]+# = &ab . (2.8)

All other commutators are zero. In order to satisfy the above commutators we canrealize "a and ca as multiplication operators and !a and ca as derivative ones:

!a = !i$

$"a, ca =

$

$ca. (2.9)

Substituting the previous relations into the Hamiltonian H of Eq. (2.5) we obtainthe following operator (from now on we will omit the hat signs ! on the symbols ofthe abstract operators. Instead we will use them on the symbols which indicate thematrices associated to these operators):

H = !i#ab$bH$a ! i#ac$c$bHcb $

$ca. (2.10)

The first term of (2.10) is just the Liouville operator that appears in the operatorialformulation of classical mechanics due to Koopman and von Neumann.2,3 Thisconfirms that (2.4) is just the correct functional counterpart of the operatorialformulation for classical mechanics. Moreover, thanks to the presence of Grassmannvariables, the CPI provides also the evolution of more generalized objects.1 In factthe following kernel:

K("f , cf , tf |"i, ci, ti) ="

D!!"D!D!!cDc exp#

i

" tf

ti

dtL$

(2.11)

●transform as a basis for vector fieldsca

●antisymmetric tensors

V (p) =1p!

V a1...ap!a1 ! · · · ! !ap "# V $ 1p!

V a1...ap ca1 · · · cap

!" functions of and ! c-10-

Global symmetries

Dynamical Symmetries(they depend on )

Simmetrie e Geometria del CPI (1)

!L = !a"a + icaca ! !a#ab$bH ! ica#ad$d$bHcb

!H = !a#ab$bH + ica#ad$d$bHcb

Trasformata di Legendre

Esistono alcune simmetrie globali e universali di !H i cuigeneratori sono i seguenti:

Simmetrie puramente

Geometriche

(Non dipendono da H("))

Algebra Isp(2)

"########$

########%

QBRS = ica!a

QBRS = ica#ab!b

K = 12#abcacb

K = 12#abcacb

Qgh = caca

Simmetrie Dinamiche

(Dipendono da H("))

"######$

######%

NH = ca$aH(")

NH = ca#ab$bH(")

QH = QBRS ! NH

QH = QBRS + NH

& '( )

Algebra N = 2 SUSY [QH , QH ] = 2i !H

4

[QH , QH ] = 2iHalgebra N=2 susy:

Simmetrie e Geometria del CPI (1)

!L = !a"a + icaca ! !a#ab$bH ! ica#ad$d$bHcb

!H = !a#ab$bH + ica#ad$d$bHcb

Trasformata di Legendre

Esistono alcune simmetrie globali e universali di !H i cuigeneratori sono i seguenti:

Simmetrie puramente

Geometriche

(Non dipendono da H("))

Algebra Isp(2)

"########$

########%

QBRS = ica!a

QBRS = ica#ab!b

K = 12#abcacb

K = 12#abcacb

Qgh = caca

Simmetrie Dinamiche

(Dipendono da H("))

"######$

######%

NH = ca$aH(")

NH = ca#ab$bH(")

QH = QBRS !NH

QH = QBRS + NH

& '( )

Algebra N = 2 SUSY [QH , QH ] = 2i !H

4

(conserved charges for any system)Universal Symmetries

K =12!abc

acb; K =12!abcacb

QBRS ! ica!a; QBRS ! ica"ab!b; Qg ! caca

Square roots of the Hamiltonian:

Q2(1) = Q2

(2) = !iH

If we apply twice the transformations generated by we get a time translation.

Q(i)

Q(1) ! QBRS " NH , Q(2) ! QBRS + NH

-11-

CARTAN CALCULUS

!"Differential Geometry Operations

Path Integral Corrispondent

!"dF (p) [QBRS, F (p)]exterior derivative

!"!V F (p)[V , F (p)]

interior contraction

!" HLie derivative

Lh = d!h + !hd

!" [QBRS,H] = 0[d,Lh] = 0

!"equivariant exterior ! derivative

Q(1)deq = d ! !h

-12-

The Way Out ?.● !

OBSERVABLESin KvN theory

● KvN did not specify which are the observables in their formulation

In standard classical mechanics the observables are only:

●

So all the observables commutes and there is no interpherence effects.

!O(!)

● If the observables were hermitian operators but functions also of , ,

then they would not commute and interpherence effects would appear

because superposition is admitted in KvN

! c c

!O(!,", c, c)

-13-

Superspace and Superfields

superspacet!

!

t !" (t, !, !)!

●(!, c, c,") target space

Superfield :!a(t, !, !) = "a(t) + !ca(t) + !#abcb(t) + i!!#ab$b(t)

!

base spacet●

-14-

Confronto Tra Path Integral

Ampiezza di transizione classica:

!!qf ; tf |!q

i ; ti" =!

D!!!qD!p exp i

!idtd!d! L[!]

Ampiezza di transizione quantistica:

!qf ; tf |qi; ti" =!

D!!qDp expi

!

!dtL["]

Il path integral classico ha la stessa Lagrangiana e lastessa misura del path integral quantistico con i super-campi al posto dei campi e il supertempo al posto deltempo.

8

Quantum transition amplitude (QPI):

Confronto Tra Path Integral

Ampiezza di transizione classica:

!!qf ; tf |!q

i ; ti" =!

D!!!qD!p exp i

!idtd!d! L[!]

Ampiezza di transizione quantistica:

!qf ; tf |qi; ti" =!

D!!qDp expi

!

!dtL["]

Il path integral classico ha la stessa Lagrangiana e lastessa misura del path integral quantistico con i super-campi al posto dei campi e il supertempo al posto deltempo.

8

Classical transition amplitude (CPI) :

Comparison ofPath Integrals

The classical path integral has the same Lagrangian and the same measure as the quantum path integral but with the fields replaced by the superfields and the time integration replaced by the supertime integration.

-15-

Local universal symmetries

●!

!a ! !a + "(t)#$abcbcb ! cb " "(t)cb

!!a ! !a + "(t)#ca

ca ! ca " "(t)ca

!!a ! !a + i"(t)##$ab%b

%a ! %a " "(t)$ab!b

● The only quantity invariant under all three transformations above is .

So is invariant and the transformations turn out to be local

symmetries of our systems

!a!

L[!]d!d!

-16-

● As they are like gauge transformations,the only admissible observables are

the following ones:

which are gauge invariant.O(!a)

Heisenberg picture for and! !

● Let us go from the Heisenberg picture in , to the

Schroedinger one ! !

● e!!QBRS!QBRS !!!a(t, !, !)e!QBRS+QBRS ! = !"a(t)

● So is the Heisenberg picture,in ( , ), of

!!a(t, !, !)

!! !!a(t)

● Let us then turn the physical observables into the

Schroedinger picture form:e!!QBRS!!QBRS !OH(!a)e!Q

BRS+!QBRS

! OS = O(!a)

!O(!a)

● We get in this way the standard observables of CM!O(!a)

-17-

Super-selection rule

● We have an operator which commutes with all the observables and which is not a multiple of the

identity.

!!a

!O(!!a)

● This triggers a superselection mechanism which says that the

Hilbert space of the system is given by the eigenvariety of the superselection

operator :!!

!!a|!a0! = !a

0 |!a0!

● Another eigenvariety will be:!!a|!a

1! = !a1 |!a

1!

-18-

No-superposition in CM

● As and belongs to different Hilbert spaces we cannot superimpose

them:

|!a1!|!a

0!

|!!a! " |!a0!+ |!a

1!

|!!a! is not a physical state.

● Not all hermitian operator are observables . The non-observables

ones are all those hermitian operators which connects different superselected

sectors.Example: the Liouvillian operator:

which moves from one point to another.!L = !a"

ab#bH

-19-

Schoredinger picture in , of the KvN

waves! !

● Besides the obsevables we have to bring also the KvN-waves in the Schoeredinger picture in , !!

!!S(t, ", ") = e!QBRS+QBRS !!(t) =

= !(#a + "ca + "$abcb, c)

● Note that commutes with all the observables so the system lives also in the eigenvariety:

ca

ca|ca0! = ca0 |ca0!

● The physical states are then:! = !"|"a

0"!c|ca0" == #("# "0)#(c# c0)

-20-

Schroedinger picture

● Is !! = "(#! #0)"(c! c0)

!!S = !("a + #ca + #$abcb, ca)

of the form?

● The only manner to achieve that is to put so:c = c = 0

!!S = "(#! #0)"(c)

and this is isomorphic to:!!S = "(#! #0)

which are the physical states.

-21-

● If we had neglected the differential forms in the KvN procedure, we would have had only functions of the , . These would make up non- commuting observables and there would be no way to get rid of them via some principle but only via a postulate:

restrict the to the

!! !!O(!!, !")

O(!!, !") !O("!)

Role of , cc

● The , instead allow us to go to the superfield and have a gauge invariance which automatically selects the correct observables.

c c

-22-

Quantization and superposition

● In this framework quantization is achieved by sending (identical to “geometric quantization”).

!, ! ! 0

● The “gauge invariace” disappears if : !, ! ! 0!

!a ! !a + "(t)#ca

ca ! ca " "(t)ca

● Having no gauge invariace the superselection mechanism is not triggered anymore and we can have superposition.

-23-

Physical Meaning of the Superfield

● We would like to get CM in the CPI formulation by starting from the path integral of QM and performing a “block spin” transformation to “block spin” of phase space of size ? >> h

●points in

phase spaceblock spin in phase space

QPI CPI

● We already passed from the QPI to the CPI via:

!a !a

so maybe the represent the block spin: !a

!a(!, c, . . .) = c = side

= center of the block spin !

If the new basic variables in CM are the blocks of phase space, then we can reparametrize its internal points as we

like provided the block remain the same.

Crucial Physical Freedom that we have

The local-transformation on the components of the superfields that we have

used for the superselection rule are, “maybe”, the above Internal Point

Reparametrization

RENORMALIZATION GROUP APPROACH

Can we find an “analog” of the Gamma-function of the Callan-Symanzik renormalization group equation and its associated zeros ?

We proved that the Lagrangians in the QPI and the CPI are the same, except for the exchange field-superfield, it means there is only a field “renormalization” and not a coupling and mass renormalization. This is typical of supersymmetric field theory .

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seminario geometry-qm-2