Is the Feynman path integral complex enough? Gke Ba ar University - - PowerPoint PPT Presentation
Is the Feynman path integral complex enough? Gke Ba ar University of North Carolina, Chapel Hill 04.22.2020 [with A. Alexandru, P. Bedaque, N. Warrington, G. Ridgway] 1 Motivations first-principles studies of strongly interacting
04.22.2020
1
[with A. Alexandru, P. Bedaque, N. Warrington, G. Ridgway]
2
first-principles studies of strongly interacting systems
3
temperature 1012 K chemical potential ∼310 MeV
heavy ion coll. early universe
quark gluon plasma hadron gas
you are here neutron stars
critical point
4
first-principles studies of strongly interacting systems
5
Heavy ion collisions: Quark gluon plasma is a liquid !
6
Heavy ion collisions: Quark gluon plasma is a liquid !
7
Heavy ion collisions: Quark gluon plasma is a liquid !
8
Heavy ion collisions: Quark gluon plasma is a liquid !
9
Quark gluon plasma is a liquid what is the viscosity, conductivity …?
10
0.05 0.1 0.15 0.2 0.25 0.3
T (K) Hole doping x
~ T2 ~ T + T2
TFL? Tcoh? ~ T (T) S-shaped T* d-wave SC ~ Tn (1 < n < 2) A F M upturns in (T) [N.E. Hussey, ’08 ]
first-principles studies of strongly interacting systems
11
animation: Derek Leinweber, University of Adelaide
we are interested in expectation values
⟨n⟩ ⇔ equation of state ⟨J(t)J(0)⟩ ⇔ conductivity ⟨Tab(t)Tcd(0)⟩ ⇔ viscosity
examples:
12
~ ""g7
VO1.UME 20, NUMBER 2
Space- ..ime A~~~iroac. i 1:o .5 on-.le. .a1:ivistic
4 uantuns
.V.:ec.zanies
Cornell University,
Ithaca, Veto York
Non-relativistic quantum mechanics is formulated here in a different
mathematically equivalent
to the familiar formulation. In quantum
mechanics the probability
which can happen in several different ways is the absolute square
complex contributions,
that a particle
will be
found to have a path x(t) lying somewhere within a region of space time is the square of a sum
in the region. The contribution from a single path is
postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's equation.
The relation to matrix and operator algebra is discussed.
Applications are indicated, in particular
to eliminate
the coordinates
from the equations
electrodynamics.
;'I is a curious
historical
fact that
modern
& - quantummechanics began with two quite di8'erent mathematical formulations:
the differ-
ential equation
and the matrix algebra of Heisenberg.
The two, apparently
dis- similar approaches, were proved
to be mathe-
matically equivalent. These two points of view
were, destined
to complement
and
to be ultimately
synthesized
in Dirac's
trans- formation theory.
This paper
will describe what
is essentially
a
third formulation
quantum
was suggested
by some
concerning
the relation
classical action' to quantum
bility amplitude
is associated
with an
entire motion of a particle as a function of time, rather than
simply with a position of the particle at a
particular time.
The formulation
is mathematically
equivalent
to
the
more usual formulations.
There are,
therefore, no fundamentally
new results.
How- ever, there is a pleasure
in recognizing
from a new point of view. Also, there are prob- lems for which
the new
point
distinct advantage.
For example,
if two systems A and 8 interact,
the coordinates
systems, say 8, may
be eliminated
from
the
equations describing the motion of A. The inter-
' P. A. M. Dirac, The Principles
3Eeohanics
(The
Clarendon Press, Oxford,
1935), second
edition,
Section 33;also, Physik. Zeits. Sowjetunion 3, 64 (1933).
' P. A. M. Dirac, Rev. Mod. Phys.
1'7, 195 (1945).
3 Throughout
this paper the term "action" will be used for the time integral
Lagrangian along a path. %'hen this path is the one actually taken by a particle, moving classically,
the integral
should
more properly be called Hamilton's
6rst principle
function.
367
x(t)
space is the square of a sum
in the region. The contribution
from a single path is postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's
equation. The relation to matrix
ⅇ
ⅈ S [ x ( t ) ]
x(t)
13
x(t)
space is the square of a sum
in the region. The contribution
from a single path is postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's
equation. The relation to matrix
field field
~ ""g7
VO1.UME 20, NUMBER 2
Space- ..ime A~~~iroac. i 1:o .5 on-.le. .a1:ivistic
4 uantuns
.V.:ec.zanies
Cornell University,
Ithaca, Veto York
Non-relativistic quantum mechanics is formulated here in a different
mathematically equivalent
to the familiar formulation. In quantum
mechanics the probability
which can happen in several different ways is the absolute square
complex contributions,
that a particle
will be
found to have a path x(t) lying somewhere within a region of space time is the square of a sum
in the region. The contribution from a single path is
postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's equation.
The relation to matrix and operator algebra is discussed.
Applications are indicated, in particular
to eliminate
the coordinates
from the equations
electrodynamics.
;'I is a curious
historical
fact that
modern
& - quantummechanics began with two quite di8'erent mathematical formulations:
the differ-
ential equation
and the matrix algebra of Heisenberg.
The two, apparently
dis- similar approaches, were proved
to be mathe-
matically equivalent. These two points of view
were, destined
to complement
and
to be ultimately
synthesized
in Dirac's
trans- formation theory.
This paper
will describe what
is essentially
a
third formulation
quantum
was suggested
by some
concerning
the relation
classical action' to quantum
bility amplitude
is associated
with an
entire motion of a particle as a function of time, rather than
simply with a position of the particle at a
particular time.
The formulation
is mathematically
equivalent
to
the
more usual formulations.
There are,
therefore, no fundamentally
new results.
How- ever, there is a pleasure
in recognizing
from a new point of view. Also, there are prob- lems for which
the new
point
distinct advantage.
For example,
if two systems A and 8 interact,
the coordinates
systems, say 8, may
be eliminated
from
the
equations describing the motion of A. The inter-
' P. A. M. Dirac, The Principles
3Eeohanics
(The
Clarendon Press, Oxford,
1935), second
edition,
Section 33;also, Physik. Zeits. Sowjetunion 3, 64 (1933).
' P. A. M. Dirac, Rev. Mod. Phys.
1'7, 195 (1945).
3 Throughout
this paper the term "action" will be used for the time integral
Lagrangian along a path. %'hen this path is the one actually taken by a particle, moving classically,
the integral
should
more properly be called Hamilton's
6rst principle
function.
367
~ ""g7
VO1.UME 20, NUMBER 2
Space- ..ime A~~~iroac. i 1:o .5 on-.le. .a1:ivistic
4 uantuns
.V.:ec.zanies
Cornell University,
Ithaca, Veto York
Non-relativistic quantum mechanics is formulated here in a different
mathematically equivalent
to the familiar formulation. In quantum
mechanics the probability
which can happen in several different ways is the absolute square
complex contributions,
that a particle
will be
found to have a path x(t) lying somewhere within a region of space time is the square of a sum
in the region. The contribution from a single path is
postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's equation.
The relation to matrix and operator algebra is discussed.
Applications are indicated, in particular
to eliminate
the coordinates
from the equations
electrodynamics.
;'I is a curious
historical
fact that
modern
& - quantummechanics began with two quite di8'erent mathematical formulations:
the differ-
ential equation
and the matrix algebra of Heisenberg.
The two, apparently
dis- similar approaches, were proved
to be mathe-
matically equivalent. These two points of view
were, destined
to complement
and
to be ultimately
synthesized
in Dirac's
trans- formation theory.
This paper
will describe what
is essentially
a
third formulation
quantum
was suggested
by some
concerning
the relation
classical action' to quantum
bility amplitude
is associated
with an
entire motion of a particle as a function of time, rather than
simply with a position of the particle at a
particular time.
The formulation
is mathematically
equivalent
to
the
more usual formulations.
There are,
therefore, no fundamentally
new results.
How- ever, there is a pleasure
in recognizing
from a new point of view. Also, there are prob- lems for which
the new
point
distinct advantage.
For example,
if two systems A and 8 interact,
the coordinates
systems, say 8, may
be eliminated
from
the
equations describing the motion of A. The inter-
' P. A. M. Dirac, The Principles
3Eeohanics
(The
Clarendon Press, Oxford,
1935), second
edition,
Section 33;also, Physik. Zeits. Sowjetunion 3, 64 (1933).
' P. A. M. Dirac, Rev. Mod. Phys.
1'7, 195 (1945).
3 Throughout
this paper the term "action" will be used for the time integral
Lagrangian along a path. %'hen this path is the one actually taken by a particle, moving classically,
the integral
should
more properly be called Hamilton's
6rst principle
function.
367
Fields
all fields
14
domain of PI: space of all fields
ϕ
x(t)
space is the square of a sum
in the region. The contribution
from a single path is postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's
equation. The relation to matrix
field field
real part with imaginary time
~ ""g7
VO1.UME 20, NUMBER 2
Space- ..ime A~~~iroac. i 1:o .5 on-.le. .a1:ivistic
4 uantuns
.V.:ec.zanies
Cornell University,
Ithaca, Veto York
Non-relativistic quantum mechanics is formulated here in a different
mathematically equivalent
to the familiar formulation. In quantum
mechanics the probability
which can happen in several different ways is the absolute square
complex contributions,
that a particle
will be
found to have a path x(t) lying somewhere within a region of space time is the square of a sum
in the region. The contribution from a single path is
postulated
to be an exponential
whose (imaginary) phase is the classical action (in units of h)
for the path in question. The total contribution
from all paths reaching x, t from the past is the wave function P(x, t). This is shown to satisfy Schroedinger's equation.
The relation to matrix and operator algebra is discussed.
Applications are indicated, in particular
to eliminate
the coordinates
from the equations
electrodynamics.
;'I is a curious
historical
fact that
modern
& - quantummechanics began with two quite di8'erent mathematical formulations:
the differ-
ential equation
and the matrix algebra of Heisenberg.
The two, apparently
dis- similar approaches, were proved
to be mathe-
matically equivalent. These two points of view
were, destined
to complement
and
to be ultimately
synthesized
in Dirac's
trans- formation theory.
This paper will describe what
is essentially
a
third formulation
quantum
was suggested
by some
concerning
the relation
classical action' to quantum
bility amplitude
is associated
with an
entire motion of a particle as a function of time, rather than
simply with a position of the particle at a
particular time.
The formulation
is mathematically
equivalent
to
the
more usual formulations.
There are,
therefore, no fundamentally
new results.
How- ever, there is a pleasure
in recognizing
from a new point of view. Also, there are prob- lems for which
the new
point
distinct advantage.
For example,
if two systems A and 8 interact,
the coordinates
systems, say 8, may
be eliminated
from
the
equations describing the motion of A. The inter-
' P. A. M. Dirac, The Principles
3Eeohanics
(The
Clarendon Press, Oxford,
1935), second
edition,
Section 33;also, Physik. Zeits. Sowjetunion 3, 64 (1933).
' P. A. M. Dirac, Rev. Mod. Phys.
1'7, 195 (1945).
3 Throughout
this paper the term "action" will be used for the time integral
Lagrangian along a path. %'hen this path is the one actually taken by a particle, moving classically,
the integral
should
more properly be called Hamilton's
6rst principle
function.
367
Fields Lattice
e−i ̂
Ht → e− ̂ Hτ
thermal physics!
Main features:
15
e−i ̂
Ht → e− ̂ Hτ
thermal physics!
Main features:
H/T ̂
finite positive
e−S[ϕ]
16
e−S[ϕ]
importance of the field configuration 𝜚:
|
¥r¥
¥ A
. . . . . . . . . . . .As
"
.space of all fields
pick out the important (small action) configurations path integral ~ statistical average with
P(ϕ) ∝ e−S[ϕ]
𝒪
a=1
17
lattice importance sampling (Monte-Carlo)
18
19
In a variety of problems of interest S is complex
. . .
e−S[ϕ] is not a probability distribution
20
2 4 x
3.5x10191 Re[ⅇ-S]
∫
∞ −∞
e−(x+42i)2dx = 2 π
finite density
∝ e−SR[ϕ]
importance ∝ e−SR[ϕ]
“reweighting”
⟨𝒫⟩ = ⟨𝒫e−iSI[ϕ]⟩SR ⟨e−iSI[ϕ]⟩SR ⟨e−iSI[ϕ]⟩SR ∝ e−volume/T
need exponentially large resources
21
low − T
22
low − T
23
. . .
μ
∫
∞ −∞
e−(x+42i)2dx = 2 π
horrific sign problem
2 4 10 20 30 40 50
z
2 4
1×10191 2×10191 3×10191
24
horrific sign problem better
2 4 10 20 30 40 50
z
2 4
1×10191 2×10191 3×10191
2 4
2×1015 4×1015
25
∫𝒟 e−(z+42i)2dz = 2 π
2 4 10 20 30 40 50
z
2 4
2 4 6 8 10
2 4
1×10191 2×10191 3×10191
2 4
2×1015 4×1015
∫𝒟 e−(z+42i)2dz = 2 π
horrific sign problem better much better
26
Review article : ”Complex paths around the sign problem” [Alexandru, GB, Bedaque, Warrington] coming soon… [also work by Cristoforetti, Di Renzo et al, Fujii et al., Tanizaki et al.,… ] Mathematical origins: Picard-Lefschetz theory [Pham, Fedoryuk, Witten, ….]
27
N
.t
#
..
complex field space
deformed domain
ℝN ℝN
(“allowed”) (“good”)
28
N
.t
#
..
ϕ(0) ϕ(Tflow)
ℝN ℝN
(“allowed”) (“good”)
dϕ(τ) dτ = ∂S[ϕ] ∂ϕ follow an equation of motion, ``holomorphic gradient flow”
29
deformation discretization importance sampling
30
Dynamics Finite density
i ℏ S[ϕ]𝒫(t)𝒫(0)
i ℏ S[ϕ] leads to quantum interference
31
interacting Bose gas:
ℒ = 1 2(∂ϕ)2 − 1 2 m2ϕ2 − λ 4! ϕ4
free theory 𝝁=0
0.0 0.5 1.0 1.5 2.0
2 4 t Cp=0(t)
0.5 1.0 1.5 2.0
2 4 t Cp=2 πL(t)
Cp(t) = ⟨ϕ(t, p)ϕ(0,p)⟩β
[Alexandru, GB, Bedaque, Ridgway, Vartak, Warrington, PRL 117081602, PRD 95 114501]
32
interacting Bose gas:
ℒ = 1 2(∂ϕ)2 − 1 2 m2ϕ2 − λ 4! ϕ4
weak coupling 𝝁=0.1
free theory
0.0 0.5 1.0 1.5 2.0
2 4 t Cp=0(t)
0.5 1.0 1.5 2.0
2 4 t Cp=2 πL(t)
Cp(t) = ⟨ϕ(t, p)ϕ(0,p)⟩β
[Alexandru, GB, Bedaque, Ridgway, Vartak, Warrington, PRL 117081602, PRD 95 114501]
33
1st order perturbation free theory
0.0 0.5 1.0 1.5 2.0
2 4 t Cp=0(t)
interacting Bose gas:
ℒ = 1 2(∂ϕ)2 − 1 2 m2ϕ2 − λ 4! ϕ4
weak coupling 𝝁=0.1
0.5 1.0 1.5 2.0
2 4 t Cp=2 πL(t)
Cp(t) = ⟨ϕ(t, p)ϕ(0,p)⟩β
[Alexandru, GB, Bedaque, Ridgway, Vartak, Warrington, PRL 117081602, PRD 95 114501]
34
0.5 1.0 1.5
0.0 0.1 0.2 0.3 t Cp=2 πL(t)
0.0 0.5 1.0 1.5
2 4 t Cp=0(t)
interacting Bose gas:
ℒ = 1 2(∂ϕ)2 − 1 2 m2ϕ2 − λ 4! ϕ4
strong coupling 𝝁=1 Cp(t) = ⟨ϕ(t, p)ϕ(0,p)⟩β
[Alexandru, GB, Bedaque, Ridgway, Vartak, Warrington, PRL 117081602, PRD 95 114501]
35
0.5 1.0 1.5
0.0 0.1 0.2 0.3 t Cp=2 πL(t)
free theory
0.0 0.5 1.0 1.5
2 4 t Cp=0(t)
interacting Bose gas:
ℒ = 1 2(∂ϕ)2 − 1 2 m2ϕ2 − λ 4! ϕ4
strong coupling 𝝁=1 Cp(t) = ⟨ϕ(t, p)ϕ(0,p)⟩β
[Alexandru, GB, Bedaque, Ridgway, Vartak, Warrington, PRL 117081602, PRD 95 114501]
36
0.5 1.0 1.5
0.0 0.1 0.2 0.3 t Cp=2 πL(t)
1st order perturbation free theory
0.0 0.5 1.0 1.5
2 4 t Cp=0(t)
interacting Bose gas:
ℒ = 1 2(∂ϕ)2 − 1 2 m2ϕ2 − λ 4! ϕ4
strong coupling 𝝁=1 Cp(t) = ⟨ϕ(t, p)ϕ(0,p)⟩β
[Alexandru, GB, Bedaque, Ridgway, Vartak, Warrington, PRL 117081602, PRD 95 114501]
37
[see also follow-up by Mou, Saffin, Tranberg, ‘18]
Case Study : 0+1 d anharmonic oscillator
ℒ = 1 2 · ϕ2 − 1 2 m2ϕ2 − λ 4! ϕ4
in progress
38
p z z’ x x’
[also (finite density) Fujii, Honda, Kato, Kikukawa, Komatsu, Sano, JHEP 10 (2013) 147 01]
Case Study : 0+1 d anharmonic oscillator
ℒ = 1 2 · ϕ2 − 1 2 m2ϕ2 − λ 4! ϕ4
in progress
39
δRe<T x(t)x(0)>
1 2 3 4
0.00 0.02 0.04 t
1 2 3 4
0.0 0.5 1.0 t
Nt = 24, Nβ = 4, λ = 24
g2
chain of interacting fermions S = ∫ d2x ¯ ψa (γμ∂μ + m + μγ0) ψa + g2 2Nf (ψaγμψa)(ψbγμψb) → Nf 2g2 ∫ d2xAμAμ + tr log(∂ + A + μγ0 + m) / /
asymptotically free, sign problem at finite density [Alexandru, GB, Bedaque, Ridgway, Warrington, Phys. Rev. D95, 014502 ]
40
ℝN
1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 μ/mf Re〈e-ⅈ SI〉
ℝN
1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 μ/mf 〈n〉/mf
sign problem equation of state
[Alexandru, GB, Bedaque, Ridgway, Warrington, Phys. Rev. D95, 014502 ]
41
ℝN Tflow=0
1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 μ/mf 〈n〉/mf
ℝN Tflow=0
1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 μ/mf Re〈e-ⅈSI〉
sign problem equation of state
[Alexandru, GB, Bedaque, Ridgway, Warrington, Phys. Rev. D95, 014502 ]
42
ℝN Tflow=0 Tflow=0.4
1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 μ/mf 〈n〉/mf
ℝN Tflow=0 Tflow=0.4
1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 μ/mf Re〈e-ⅈSI〉
sign problem equation of state
[Alexandru, GB, Bedaque, Ridgway, Warrington, Phys. Rev. D95, 014502 ]
43
T/mf0.38 T/mf0.19 T/mf0.13 T/mf0.09
0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 /mf n/mf
particularly bad sign problem: ⟨e−iSI[ϕ]⟩SR ∝ e−volume/T
[Alexandru, GB, Bedaque, Ridgway, Warrington, Phys. Rev. D95, 014502 ]
44
Nt×Nx=10×10 Nt×Nx=10×20
1 2 3 4 0.0 0.5 1.0 1.5 μ/mf 〈n〉/mf
Nt×Nx=12×12 Nt×Nx=16×16 Nt×Nx=20×20
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 μ/mf 〈n〉/mf
continuum limit thermodynamic limit
[Alexandru, GB, Bedaque, Ridgway, Warrington, Phys. Rev. D95, 014502 ]
45
baryon 2
QED with 3 ``quarks” with charges q=2,-1,-1 S =
3
∑
a=1 ∫ d2x [F2 + ¯
ψ a (γμ(∂μ − gqaAμ) + m − μγ0) ψ a ]
0.2 0.4 0.6 0.8 1 1 2 3 4 5 hσi µB/mB RN
10
MT=0.01 MT=0.02 MT=0.05 0.2 0.4 0.6 0.8 1 1 2 3 4 5 hni µB/mB RN
10
MT=0.01 MT=0.02 MT=0.05
sign problem equation of state
46
[Zambello, Di Renzo, Phys. Rev. D95, 014502 ] n: density L: Polyakov loop
mq → ∞ effective theory of Polyakov loops
μ ≠ 0
ℳ ~ ∑ ``Lefschetz thimbles” (fixed points of flow+fluctuations)
47
[Ulybyshev, Winterowd, Zafeiropoulos PRD 101 (1), 014508]
48
2d Hubbard model away from half filling on a Honeycomb lattice
hcos ImSi hcos arg Ji hΣGi BSS-QMC 0.2363±0.0032 0.2363±0.0032 HMC,α=1.0 0.9627±0.0038 0.427±0.014 0.351±0.015 HMC,α=0.8 0.797±0.022 0.915±0.008 0.644±0.028
average sign conventional MC deformation ~ ∑ “thimbles”
(1a)
fixed point of flow = saddle point of S[𝜚]
[Alexandru, Bedaque, Lamm, Lawrence Phys.Rev.D 96 (2017) 9, 094505] ϕ Im ˜ ϕ
N
.t
#
..
ϕ ˜ ϕ
49
[Mori et al. ’17-’19, Alexandru et al. ‘18, Bursa et al. ’18, Kashiwa et al. ‘19, Detmold et al. ‘20]
⟨e−iSI⟩λ = ∫ℳλ d[ϕ]e−S ∫ℳλ d[ϕ]e−SR maximize the average phase: within a family of manifolds minimize the sign problem ℳλ
N
.t
#
..
50
??
critical pointflow QFT: real time dynamics, finite density
simulated annealing
φ0 φ0 φ1 φ0
1φn−1 φ0
n1φn ... ... accept/reject
Metropolis step w/ p1 Metropolis step w/ pn Metropolis step w/ pn Metropolis step w/ p1
Hybrid Monte Carlo pseudo-fermions, estimators machine learning, path optimization
51