![Page 1: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/1.jpg)
Wigner Functional Methodin
Quantum Field Theory
Berndt MüllerYIPQS Molecule: “Entropy Production”
YITP Kyoto - 13 August 2008
Based on: S. Mrówczyński & BM, PRD 50, 7542 (1994)
![Page 2: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/2.jpg)
An impossible dream?
2
Simulations of real-time dynamics of quantum fields far off equilibrium have used a variety of approximation schemes:
Boltzmann equations for (quasi-)particle excitations:
Good in dilute systems, but loss of coherence and off-shell effects.
Classical nonlinear (lattice) field equations:
Good for highly occupied modes, but loss of all quantum effects.
Real-time Green function techniques:
Practical for homogeneous or low-dimensional systems only.
Is it possible to devise a scheme which encompasses the classical field and particle limit (useful for phenomenology), yet includes essential quantum effects, such as the uncertainty relation, phases, or correlations, yet can be treated by similar methods similar to those that have already been developed?
![Page 3: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/3.jpg)
W (q, p; t) =!
du e!ipu!q +12u| !(t) |q " 1
2u#
W (q, p) = W (q, p)!!dqdp
2!W (q, p) = Tr " = 1
!W
!t= ! p
m
!W
!q! i
!
!!V
!q
"q +
i!2
!
!p
#! !V
!q
"q ! i!
2!
!p
#$W
!!0!" ! p
m
!W
!q+
!V
!q
!W
!p
Wigner function
3
Reminder - Usual definition of Wigner function for a point particle:
Properties:
Equation of motion (for H = p²/m + V) :
![Page 4: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/4.jpg)
Tr !2 = Tr ! = 1 !!
dqdp
2"W (q, p)2 =
!dqdp
2"W (q, p) = 1
!P,a(q) =!
2"/! eiPq!!(q!a)2
W (q, p) = 2 exp!!2!(q ! a)2 ! 1
2!(p! P )2
"
!(q " a)2# =1
4!!(p " P )2# = !
(!(!q)2"!(!P )2"1/2 =12
Wave packets
4
Consider a Gaussian wavepacket:
The Wigner transform is:
Pure quantum state:
Δ→∞: Good position
Δ→0: Good momentum
Minumum uncertainty wavepacket
![Page 5: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/5.jpg)
Wigner functional
5
W [!(x),"(x); t] =!
D!(x) exp"!i
!dx "(x)!(x)
#"!(x) +
12!(x)| "(t) |!(x) ! 1
2!(x)#
W [!(p),"(p); t] =!
D!(p) exp"!i
! !
0dp
#""(p)!(p) + "(p)!"(p)
$%"!(p) +
12!(p)| "(t) |!(p) ! 1
2!(p)#
Wigner functional = Adaptation to QFT. Start with a scalar quantum field Φ.
Momentum space representation:
Adapt quantum phase space formulation (Wigner function) to the field representation of quantum field theory in order make the classical field limit more transparent.
Position space representation:
![Page 6: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/6.jpg)
! D!2!
exp"±i
!dx !(x)"(x)
#= #("(x))
Z ! Tr ! =!
D!(x)"!(x)| ! |!(x)# =!
D!(x)D"(x)
2"W [!,"; t]
!O(!, ")" =1Z
Tr!!(t)O(!, ")
"=
1Z
#D!(x)
D"(x)2"
O(!,") W [!,"; t]
Basics
6
Using it is obvious that
W [!(x),"(x); t] =!
D!(x) exp"!i
!dx "(x)!(x)
#"!(x) +
12!(x)| "(t) |!(x) ! 1
2!(x)#
implies
Similarly, by inserting complete sets of states and partial integrations:
Wigner functional expression corresponds to symmetrized quantum operators.
![Page 7: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/7.jpg)
i! !
!t"(t) = [H, "(t)] H =
12
!dx
"!2(x) +
#!"(x)
$2 + m2"2(x)" LI(x)%
!
!tW [!,"; t] =
!D" exp
"! i
!
!dx "(x)"(x)
#"! +
12"| [H, #] |!! 1
2"#
$ G! + G! + Gm + GI
Gm =m2
2
!dx
!D! exp
"! i
!
!dx !(x)!(x)
#
"$%
"(x) +12!(x)
&2 !%"(x)! 1
2!(x)&2
' (" + 1
2!| " |"! 12!
)
Gm = i!m2
!dx !(x)
!
!"(x)W [!,"; t]
Equation of motion - I
7
![Page 8: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/8.jpg)
!!
!t+
"dx
#!(x)
"
""(x)!
$m2"(x)!"2"(x)
% "
"!(x)+KI(x)
&'W [",!; t] = 0
!
!tW [!,"; t] = !
!dx
""H
""(x)"
"!(x)! "H
"!(x)"
""(x)+ O
"!2 "3
""3
##W [!,"; t]
Equation of motion - II
8
KI(x) ! " i
!LI
!!(x) +
i!2
!
!"(x)
"+
i
!LI
!!(x)" i!
2!
!"(x)
"
LI(!) = ! !
4!!4 !" KI =
!
6
!!!3(x)
"
""(x)+
!2
4!(x)
"3
""3(x)
"
Other terms are derived similarly, final result is:
with
Standard example: Φ⁴ theory:
Thus, evolution of W is described by a classical transport equation with quantum corrections:
![Page 9: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/9.jpg)
H0 =! !
0
dp
2!
"!†(p)!(p) + (p2 + m2)"†(p)"(p)
#! =
1Z
e!!H0
!W! [!,"] = C exp"!1
2!
# !
"!
dp
2"#!(p)
$"#(p)"(p) + E2(p)!#(p)!(p)
%&
!!(p) =2
!E(p)th
!E(p)2
C ! exp!" !
"!
dp
2!ln th
"E(p)2
#
! ! 0 : !!(p)! 1.
W! [!(x),"(x)] = C ! exp!!1
2!
"dx dx! #(x! x!)
#"(x)"(x!) +"!(x)"!(x!) + m2!(x)!(x!)
$%
Free field at T ≠ 0
9
Densitymatrix: with
Momentum space Wigner functional (ħ = 1):
with
(classical limit)
Coordinate space form contains “nonlocal” Hamiltonian:
![Page 10: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/10.jpg)
!!†(p)!(p)" =1
2 E(p) th!E(p)2
# 12 E(p)
!"†(p)"(p)" =E(p)
2 th!E(p)2
# E(p)2
!!!†(p)!(p)" !"†(p)"(p)" =
12 th!E(p)
2
# 12
Fluctuations
10
Thermal quantum fluctuations are larger than classical fluctuations:
implying the uncertainty relation:
![Page 11: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/11.jpg)
Z[j(x)] !!D!
D"2!
W [!,"] exp"
"
!dx !(x)j(x)
#
=!D!
D"2!
exp$""
!dx dx!
%H(x, x!)" #(x" x!)!(x)j(x!)
&'
!!(x)!(y)" =1!2
1Z[j]
"2Z[j]"j(y)"j(x)
!!!!!j(x)=0
Z0[j(x)] = N exp!!
2
"dx dx! j(x)G(x! x!)j(x!)
#
G(x) =!
dp
2!
e!ipx
!!(p)"p2 + m2
# m"0!" 14!|x|
sh(2!|x|/")ch(2!|x|/")! 1
Generating functional
11
Perturbation theory is based on generating functional
Two point function:
Free field:
with
![Page 12: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/12.jpg)
! = a†(p)|0!"0|a(p)
W1[!(p),"(p)] = C
!2
|"(p)|2 + E(p)2!(p)|2
E(p)! 1
"exp
!! |"(p)|2 + E(p)2!(p)|2
E(p)
"
! = (1 ! f(p))|0"#0| + f(p)a†(p)|0"#0|a(p)
Wf [!(p),"(p)] = C
!1 + 2f(p)
"|"(p)|2 + E(p)2!(p)|2
E(p)! 1
#$exp
"! |"(p)|2 + E(p)2!(p)|2
E(p)
#
" C ! exp"! |"(p)|2 + E(p)2!(p)|2
[1 + 2f(p)]E(p)
#
Particles
12
Consider one-particle state:
Wigner functional:
Partially occupied state:
Wigner functional:
Agrees with thermal expression for f(p) = (e-βE(p)-1)-1.
![Page 13: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/13.jpg)
m2(t) = m2 !(!t)! µ2 !(t)
H(t) =p2
2+
m(t)2
2x2
Roll-over transition
13
Decay of an unstable “vacuum state” is a common problem e.g. in cosmology (end of inflation) or condensed matter physics (phase transitions). Paradigm case: “roll-over”.
t = 0 t = 1 t = 2
t < 0 |Ψ(x)|²
|Ψ(x)|² |Ψ(x)|² |Ψ(x)|²
V(x)
V(x)V(x)V(x)
with
![Page 14: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/14.jpg)
Wigner function
14
t = 0
t = 2t = 1
t = 0.5
p
x
![Page 15: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/15.jpg)
H(t) =! !
0
dp
2!
"!†(p)!(p) + (m2(t) + p2) "†(p)"(p)
#
m2(t) = m2 !(!t)! µ2 !(t)
H(t > 0) =! µ
0
dp
2!
"!†(p)!(p)! "2
!(p) "†(p)"(p)#
+! "
µ
dp
2!
"!†(p)!(p) + "2
+(p) "†(p)"(p)#
!±(p) !!±(p2 " µ2) is always real
!W [!,", t < 0] = C exp"!1
2!
# !
"!
dp
2"#!(p)
$"#(p)"(p) + E2(p)!#(p)!(p)
%&
Roll-over in QFT
15
Initial state (not necessarily vacuum):
Hamiltonian for t > 0:
![Page 16: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/16.jpg)
!(p, t) = !(p) ch(!!(p)t)! 1!!(p)
"(p) sh(!!(p)t)
"(p, t) = "(p) ch(!!(p)t)! !!(p) !(p) sh(!!(p)t)
!(p, t) = !(p) cos(!+(p)t)! 1!+(p)
"(p) sin(!+(p)t)
"(p, t) = "(p) cos(!+(p)t) + !+(p) !(p) sin(!+(p)t)
0 < p < µ
p > µ
W [!(p),"(p); t] = W [!(p,!t),"(p,!t); 0]
!W [!,"; t] = C exp"!1
2!
# !
"!
dp
2"#!(p)
$"#(p,!t)"(p,!t) + E2(p)!#(p,!t)!(p,!t)
%&
d
dt!(p, t)± !2
±(p) "(p, t) = 0
d
dt"(p, t) = !(p, t)
!(p, 0) = !(p),
"(p, 0) = "(p)
Solution
16
Classical field trajectories:
with
![Page 17: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/17.jpg)
!W [!; t] !" D!
2!!W [!,"; t] = C !exp
#""
2
""0
dp2! #"(p) !(p, t) !#(p)!(p)
$
!(p, t) = !(µ! p)E2(p)"2
!(p)"2!(p) ch2("!(p)t) + E2(p) sh2("!(p)t)
+!(p! µ)E2(p)"2
+(p)"2
+(p) cos2("+(p)t) + E2(p) sin2("+(p)t)
!!†(p)!(p)"t = !!†(p)!(p)"0!ch2(!!(p)t) +
E2(p)!2!(p)
sh2(!!(p)t)"
G(x, t) =!
dp
2!
e!ipx
!!(p) "(p, t)t"#!" #mµ
64 th (#m/2)
"1 +
µ2
m2
#(!µt)!3/2 exp
$2µt! µx2
4t
%
Field correlator
17
Distribution of field amplitudes:
with
Field fluctuations:
Field correlation function:
![Page 18: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/18.jpg)
!
4!!4(x) ! !
4"!2(x)# !2(x)
!!2(x)" =1Z
!D!(x)
D"(x)2!
!2(x) W [!,"; t]
m2! = m2 +
!
2!!2(x)"
m2! = m2 +
!
2G(0)
= m2ren +
!
2
! "
#"
dp
2"
"p2 + m2
!
exp##"
p2 + m2!$! 1
1p2 + m2
!
Mean field approximation
18
Reduce nonlinear interactions by means of expectation values:
Self-consistentequation for m*:
![Page 19: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/19.jpg)
!Kc + !2Kq) W = 0
KcWc = 0
W = Wc + !2Wq
KcWq = !KqWc +O(!2)
Wq(q, p; t) =!
dt!dq!dp! Gc(q, q!, p, p!; t, t!) KqWc(q!, p!; t!)
KcGc(q, q!, p, p!; t, t!) ! !Gc
!t+
!H
!p
!Gc
!q" !H
!q
!Gc
!p= "(t" t!) "(q " q!) "(p" p!)
Gc(q, q!, p, p!; t, t!) = !(t! t!) "!q ! q(t)! q! + q(t!)
""!p! p(t)! p! + p(t!)
"
Semi-classical approx.
19
with
Transport equation:
Thus:
Classical Green function satisfies:
Solution is simply:
![Page 20: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/20.jpg)
H =g2
a
!"
l
12Ea
l Eal +
4g4
Re"
p
(1! trUp)
#
ag2H ! H, t/a! t, g2E ! E, g2!! !
! [(A(x)] =1!N
exp
!"d3x
#
a
$"b(x)
8(Aa(x)"Aa
0(x))2 +i
!E0(x)aAa(x)%&
![Ul] =!
l
!l(Ul) =!
l
1!Nl
exp"
bl
2tr(UlU
!1l0 )" 1
! tr(El0UlU!1l0 )
#
Non-abelian gauge field
20
Idea: Use generalized Gaussian wave packets to represent Wigner function in the link representation of lattice gauge fields [Gong, BM, Biró, Nucl. Phys. A568 (1994) 727].
Lattice (KS) Hamiltonian:
can be scaled to dimensionless variables:
Wave packet ansatz:
Continuum limit:
![Page 21: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/21.jpg)
He! =!
l
"12(1! f(vl)
2vl)E2
l0 + !2 3f(vl)16vl
|bl|2#
+ 4!
p
(1! fp Up0)
f(v) = I2(2v)/I1(2v) and fp =!
l!p
f(vl)
![Ul] =!
l
!l(Ul) =!
l
1!Nl
exp"
bl
2tr(UlU
!1l0 )" 1
! tr(El0UlU!1l0 )
#
Time evolution
21
!
! t2
t1!!|(i! "t " H)|!# = 0 [Ea
l , Abm] = !i!!ab!lm
Semiclassical transport equation for Wigner functional is realized by variationalprinciple for wave packets. ħ then enters in two distinct ways:
and
Variation is performed with respect to parameters: bl (t), Ul0 (t), El0 (t).
Write b = v + iw ; then express H in terms of parameters:
where
Limits: ħ→∞ ⇒ vl → 0 i.e. wave packet covers all SU(2); ħ→0 ⇒ vl → ∞ .
![Page 22: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/22.jpg)
dEal
dt=
i
f(vl)
!
p(l)
(fp!aUp)!
3!8
wl
vlEa
l ,
dUl
dt=
i
2
"1
f(vl)! 1
2vl
#ElUl,
dvl
dt=
3!8
f(vl)f !(vl)
wl
vl,
dwl
dt=
"1
f(vl)! 1
2vl
#E2
l +E2
l
4vl+
2f(vl)
!
p(l)
fpUp !316
!2
"vl +
w2l
vl
#,
! f(vl)f !(vl)
"E2
l
4v2l
+316
!2
"1! w2
l
v2l
##,
dEal
dt= i
!
p(l)
(!aUp) ,dUl
dt= iElUl.
! ! g2! = 4!"s.
Eqs. of motion
22
ħ → 0, vl → ∞ ⇒ classical limit:
Reminder:
![Page 23: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/23.jpg)
Results - 1
23
Run simulation from randomly chosen initial conditions (but identical width bl) until the system thermalizes. Measure T from electric energy distribution and determine maximal Lyapunov exponent. Compare with classical limit. Fit gives:〈v〉≈ 7/ħ.
![Page 24: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/24.jpg)
! T ! "! !v" E
0.1 0.76 0.255 0.01 70 2.130.3 0.64 0.24 0.12 23 2.480.3 0.73 0.3 0.22 22 2.920.3 0.85 0.34 0.2 21 3.310.3 0.85 0.36 0.27 21 3.250.3 0.85 0.34 0.2 22 3.250.3 0.9 0.41 0.37 20 3.740.7 0.5 0.19 0.16 9.0 3.240.7 0.8 0.41 0.54 7.4 4.40
!(T, !) = !c(T )(1 + "!)
Results - 2
24
δλ > 0 and grows with ħ !
![Page 25: Wigner Functional Method in Quantum Field Theorymuller/Talks/YITP_080813.pdf · Wigner Functional Method in Quantum Field Theory Berndt Müller YIPQS Molecule: “Entropy Production”](https://reader034.vdocument.in/reader034/viewer/2022042805/5f63fb50224f0d331a231e18/html5/thumbnails/25.jpg)
Conclusions
25
Wigner functional method provides a practical method to simulate quantum corrections to real-time evolution of fields in quantum field theory. Various approximation methods are available:
• Gaussian wave packets (with dynamical width)• Green function techniques• Mean field approximation
WF method permits smooth interpolation between field eigenstates and particle excitations. This allows for the approximate treatment of quantum coherence and uncertainty relation effects.
WF method has been applied to SU(2) gauge field with “Gaussian” SU(2) wave packets and variational time evolution. Results of a limited exploration suggest that λmax grows with ħ = 4παs.
Generalization to fermions, using techniques of AMD or FMD, possible?