On exact-WKB, resurgent structure, and the quantization conditions (arXiv:2008.00379 [hep-th]) Mithat ¨ Naohisa Sueishi 1 Syo Kamata 2 Tatsuhiro Misumi 3 Unsal 4 1 Nagoya University 2 Jiangxi Normal University 3 Akita University, Keio University 4 North Carolina State University
Introduction Introduction There are two resurgence in physics. 1. P-NP relation (for transseries) � D xe − S [ x ] Z ( � ) = (1) � periodic a n � n + e − S 1 b n � n + e − S 2 c n � n + ... � � � = (2) � � n n n The series is not converge, but asymptotic. The Borel ambiguity derived from the perturbative series has nonperturbative information. �� � ∝ ± ie − S 1 a n � n ( S + − S − ) (3) � n 2 / 55
2. Exact WKB method (for differential equation) − � 2 d 2 � � dx 2 + V ( x ) ψ ( x ) = Eψ ( x ) . (4) 2 � ψ n ( x ) � n ψ ( x ) = (5) n Then ψ ( x ) is asymptotic series. If we consider its Borel summation and its analytic continuation, ψ + I ( x ) → ψ + II ( x ) + ψ − II ( x ) (6) Riemann-Hilbert problem of the differential equation. 3 / 55
How are they related each other? 4 / 55
fundamental problems Other fundamental problems: 1. The relation among several quantization methods: • Bohr-Sommerfeld quantization • Schr¨ odinger eq. • path integral • Gutzwiller trace formula e.g. Can we derive path integral from Bohr-Sommerfeld? 2. How to determine the intersection number of Lefschetz thimble (relevant saddle points) 5 / 55
The all questions are solved. 6 / 55
Contents Introduction Exact WKB Example: Harmonic Oscillator Resolvent method Gutzwiller trace formula Maslov index Double well potential DDP(Delabaere-Dillinger-Pham) formula Double well in Gutzwiller’s form Partition function QMI(quasi-moduli integral) form The intersection number of Lefschetz thimble Summary 7 / 55
Exact WKB 8 / 55
Exact WKB In WKB analysis, we consider the ansatz given by − � 2 d 2 � � dx 2 + V ( x ) ψ ( x ) = Eψ ( x ) . (7) 2 � x S ( x,η ) dx , ψ ( x, � ) = e (8) S ( x, � ) = � − 1 S − 1 ( x ) + S 0 ( x ) + � S 1 ( x ) + � 2 S 2 ( x ) + ... (9) = S odd + S even (10) Then Schr¨ odinger eq. becomes Riccati eq. S ( x ) 2 + ∂S ∂x = � − 2 Q ( x ) , (11) � where Q ( x ) = S − 1 = 2( V ( x ) − E ) . Also we can show S even = − 1 ∂ ∂x log S odd . (12) 2 9 / 55
Therefore the WKB wave function is expressed as � x S ± dx = 1 � x a S odd dx ψ ± e ± a ( x ) = e √ S odd (13) At the leading order, this expression becomes usual WKB approximation: 1 � x Q ( x ) 1 / 4 e ± 1 ψ ± a � a ( x ) ∼ Q ( x ) dx , (14) � Now, we take Borel summation of ψ ± a ( x ) . 10 / 55
The posistion of Borel singularity depends on x . ( ψ ( x ) = � a n ( x ) � n ) → Stokes curve tells where the Stokes phenomena happens. 11 / 55
Stokes curve is defined as � x Im 1 � Q ( x ) dx = 0 (15) � a (16) � ( Q ( a ) = 2( V ( x ) − E ) x = a = 0 i.e. turning point) � x − II α + I − Figure 1: Airy: V ( x ) = x , across anti-clockwisely ψ + a, I = ψ + a, II + iψ − a, II ψ − a, I = ψ − a, II 12 / 55
When a wavefunction crosses a Stokes curve, its Stokes phenomena can be expressed as � � � � ψ + ψ + a, I a, II = M , (17) ψ − ψ − a, I a, II where the the matrix M is given by 1 i =: M + for anti-clockwisely, + 0 1 1 − i =: M − 1 for clockwisely, + + 0 1 M = (18) 1 0 =: M − for anti-clockwisely, − i 1 1 0 =: M − 1 for clockwisely, − − − i 1 13 / 55
If one considers two wave functions normalized at the different turning points: a 1 , a 2 . They are related by � a 2 a 1 ( x ) = e ± a 1 S odd ψ ± ψ ± a 2 ( x ) (19) Therefore � a 2 e + a 1 S odd � � � � � � ψ + ψ + a 1 ( x ) a 2 ( x ) 0 = N a 1 a 2 , N a 1 a 2 = . � a 2 ψ − ψ − e − a 1 S odd a 1 ( x ) a 2 ( x ) 0 (20) N is called Voros multiplier. Actually we can derive the eigenvalues and also the partition function with these tools without solving Schr¨ odinger eq. 14 / 55
Example: Harmonic Oscillator 15 / 55
Let the potential as V ( x ) = 1 2 ω 2 x 2 . Then its Stokes curve looks as follows ( E > 0 ): x + + a 1 a 2 − − II I III + + Figure 2: √ √ 2 E 2 E where a 1 = − ω , a 2 = are turning points. The blue line is ω the path of analytic continuation. 16 / 55
First, � � � � ψ + ψ + a 1 , I ( x ) a 1 , II ( x ) = M + (21) ψ − ψ − a 1 , I ( x ) a 1 , II ( x ) Second, � � � � ψ + ψ + a 1 , II ( x ) a 2 , II ( x ) = N a 1 a 2 (22) ψ − ψ − a 1 , II ( x ) a 2 , II ( x ) Then � � � � ψ + ψ + a 2 , II ( x ) a 2 , III ( x ) = M + (23) ψ − ψ − a 2 , II ( x ) a 2 , III ( x ) � a 2 A S odd = e 2 a 1 S odd ) � After all ( A = e � � � � ψ + ψ + a 1 , I ( x ) a 1 , III ( x ) = M + N a 1 a 2 M + N a 2 a 1 (24) ψ − ψ − a 1 , I ( x ) a 1 , III ( x ) � � ψ + a 1 , III ( x ) + i (1 + A ) ψ − a 1 , III ( x ) = (25) ψ − a 1 , III ( x ) 17 / 55
therefore we obtain D = 1 + A = 0 (26) This is equivalent to S odd = n + 1 � with n ∈ Z (27) 2 A In the case of harmonic oscillator, we can show � S odd = 1 � 2( V ( x ) − E ) dx = − 2 πi E � (28) � ω � A A (The higher orders of S odd don’t contribute to this integral) Therefore D ( E ) = 0 gives � � n + 1 E = � ω (29) 2 (30) the condition of Stokes curve: E > 0 determines n = 0 , 1 , 2 ... 18 / 55
Resolvent method - bridge exact WKB to the partition function and Gutzwiller 19 / 55
Resolvent method We can regard the quantization condition derived from exact WKB: � � ˆ D ( E ) = 0 as the Fredholm determinant: D = det H − E . 1 For the trace of resolvent: G ( E ) = tr H − E , it can be expressed as − ∂ ∂E log D = G ( E ) . Also � ∞ Z ( β ) e βE dβ G ( E ) = (31) 0 � ǫ + i ∞ 1 G ( E ) e − βE dE , Z ( β ) = (32) 2 πi ǫ − i ∞ where Z ( β ) = tr e − βH 20 / 55
Indeed, D = 1 + A = 1 + e − 2 πi E (33) � ω � E � � ω + 1 �� = e − πi E � ω 2 sin π (34) 2 2 π = e − πi E (35) � ω Γ( 1 2 + E � ω )Γ( 1 2 − E � ω ) G ( E ) = − ∂ ∂E log(1 + A ) (36) � � = − ∂ 2 π e − πi E ∂E log (37) � ω Γ( 1 2 + E � ω )Γ( 1 2 − E � ω ) 21 / 55
The partition function is � ǫ + i ∞ � � 1 − ∂ 2 π e − πi E e − βE dE Z = ∂E log � ω Γ( 1 2 + E � ω )Γ( 1 2 − E 2 πi � ω ) ǫ − i ∞ (38) ∞ e − β � ω ( n + 1 2 ) � = (39) n =0 Remark: We don’t have to solve the Schr¨ odinger eq. or path integral to derive the partition function. C � − 3 − 1 1 3 … … 2 2 2 2 Figure 3: C is the integration contour 22 / 55
Also, ∂E log(1 + A ) = − ∂ G ( E ) = − ∂ ∂E A (40) 1 + A ∞ = i in � � A pdx ( − 1) n , Te (41) � � n =1 (where T is the period of harmonic oscillator) This is actually the Gutzwiller trace formula of harmonic oscillator. 23 / 55
Gutzwiller trace formula 24 / 55
Gutzwiller trace formula Z ( T ) = tr e − iHT (42) � D x e iS = (43) periodic � ∞ 1 Z ( T ) e ( iE − ǫ ) T dT = − i tr G ( E ) = (44) H − E 0 therefore � ∞ 1 � D x e iS + iET G ( E ) = − i tr H − E = dT (45) 0 periodic � ∞ � D x e i Γ , = dT (46) 0 periodic 25 / 55
where Γ = S + ET . Action, S can be written as � T � S = p ˙ xdt − Hdt (47) � T � = pdx − Hdt (48) Evaluate T integral by stationary phase method d T = d S dΓ d T + E (49) d � Using pdx = 0 , d T d T = d S dΓ d T + E = − H + E (50) 26 / 55
The leading contributions are periodic classical solutions whose � � energy is E . There are n-times periodic orbit too: pdx → n pdx . � � � � Γ = S + ET = n pdx − ET + ET = n pdx ( n = 1 , 2 , 3 ... ) (51) Finally, ∞ � e in p.p.o. pdx � � G ( E ) = (52) p.p.o. n =1 p.p.o. stands for prime periodic orbit , which is a topologically distinguishable orbit among the countless periodic orbits. If we consider sub-leading term of stationary phase approximation, ∞ � − 1 / 2 δ 2 S � � � � � � T ( E ) e in p.p.o. pdx ( − 1) n � � G ( E ) ≃ i det � � δx i δx j � � p.p.o. n =1 (53) where i ( − 1) n is the Maslov index . 27 / 55
Maslov index Maslov index is the index determined by the number of negative eigenvalue of M , where M = δ 2 S = − d 2 � d t 2 − V ′′ ( x cl ) . � (54) � δxδx � x = x cl √ α = ν � | det M | e iαπ , det M = 2 . (55) Here, α is called the Maslov index . ( ν is the number of negative eigenvalues of M ) The determinant of the n -cycle is given by √ | det M | ( − 1) n . � det M = − i (56) Because the operator M has 2 n − 1 negative eigenvalues for n-cycle orbit. (and we call this ( − 1) n as Maslov index from here) 28 / 55
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