GLYPH<23>$_{1}$ ( GLYPH<2>$_{0}$ ) = GLYPH<0> 1 X m ; n = GLYPH<0>1 im GLYPH<14>$_{mp}$$_{;}$$_{nq}$ V$_{m}$$_{;}$$_{n}$e im GLYPH<2>$_{0}$ ; (23) GLYPH<18> (1) m ; n = im ¯ GLYPH<14>$_{mp}$$_{;}$$_{nq}$ V$_{m}$$_{;}$$_{n}$ ( m GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n ) 2 ; (24) GLYPH<18> (2) m ; n = i ¯ GLYPH<14>$_{mp}$$_{;}$$_{nq}$ ( m GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n ) 2 0 X m $^{0}$; n 0 N GLYPH<18> m ; m $^{0}$; n ; n 0 ( m 0 GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n $^{0}$) 2 GLYPH<2> V$_{m}$$_{+}$$_{m}$ $^{0}$; n + n 0 V GLYPH<3> m $^{0}$; n 0 ; (17) GLYPH<23>$_{2}$ = GLYPH<0> i X m ; n GLYPH<14>$_{mp}$$_{;}$$_{nq}$ 0 X m $^{0}$; n 0 N GLYPH<2> m ; m $^{0}$; n ; n 0 ( m 0 GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n $^{0}$) 2 GLYPH<2> V$_{m}$$_{+}$$_{m}$ $^{0}$; n + n 0 V GLYPH<3> m $^{0}$; n 0 e im GLYPH<2>$_{0}$ ; (25) GLYPH<18> (2) m ; n = i ¯ GLYPH<14>$_{mp}$$_{;}$$_{nq}$ ( m GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n ) 2 0 X m $^{0}$; n 0 N GLYPH<2> m ; m $^{0}$; n ; n 0 ( m 0 GLYPH<19> -$_{p}$$_{;}$$_{q}$ GLYPH<0> n $^{0}$) 2 GLYPH<2> V$_{m}$$_{+}$$_{m}$ $^{0}$; n + n 0 V GLYPH<3> m $^{0}$; n 0 ; (26) D GLYPH<2> DT = GLYPH<0> GLYPH<14> S GLYPH<14> GLYPH<2> GLYPH<17> GLYPH<0> #$_{GLYPH<2>}$ GLYPH<14> S GLYPH<14>GLYPH<18> ; (18) GLYPH<14> S GLYPH<14> GLYPH<2> = GLYPH<23> ( GLYPH<2>$_{0}$ ) : (19) D GLYPH<2> D GLYPH<2>$_{0}$ = GLYPH<0> T $^{0}$( GLYPH<2>$_{0}$ ) GLYPH<23> ( GLYPH<2>$_{0}$ ) = 1 ; (20) GLYPH<2> ( GLYPH<16> j GLYPH<2>$_{0}$ ) = GLYPH<2>$_{0}$ + GLYPH<19> - GLYPH<16> ; (21) $^{ }$dL$_{˙}$ GLYPH<18> d GLYPH<16> GLYPH<0> L$_{GLYPH<18>}$$^{!}$ #$_{GLYPH<2>}$ + GLYPH<23> ( GLYPH<2>$_{0}$ ) = 0 : (22) 8. Conclusion 7. GLYPH<2> ghost surface formulation #$_{GLYPH<2>}$ = 1 + $^{P}$m ; $_{n}$im GLYPH<18>$_{m}$$_{;}$$_{n}$ exp[ im GLYPH<2>$_{0}$ + i ( m GLYPH<19> - GLYPH<0> n ) GLYPH<16> ]. Inserting these into Eq. (22), and equating the LHS to zero at O ( GLYPH<15> ), we get and Explicit expressions for GLYPH<23>$_{1}$ , GLYPH<22>$_{2}$ , GLYPH<18> (1) m ; $_{n}$; GLYPH<18> (2) m ; n and GLYPH<2> (1) m ; n are given in Ref. 4, but, as there is a misprint in GLYPH<18> (2) m ; $_{n}$, we give below a corrected version, where N GLYPH<18> m ; m $^{0}$; n ; n 0 GLYPH<17> m $^{0}$( m + m $^{0}$) $^{h}$m + m 0 + m $^{0}$GLYPH<14>$_{(}$$_{m}$$_{+}$$_{m}$ $^{0}$) p ; ( n + n $^{0}$) q$^{i}$ . Also, ¯ GLYPH<14>$_{mp}$$_{;}$$_{nq}$ GLYPH<17> 1 GLYPH<0> GLYPH<14>$_{mp}$$_{;}$$_{nq}$ , where GLYPH<14>$_{mp}$$_{;}$$_{nq}$ GLYPH<17> 1 when mp = nq , is 0 otherwise, and the prime on the sum over m 0 and n 0 indicates that the resonant terms, m $^{0}$p = n $^{0}$q , are to be deleted. which are the same results as in Ref. 4, showing that GLYPH<18> and GLYPH<2> ghost orbits (and, by definition, reconciled QFMin orbits) are equivalent at this order. At O ( GLYPH<15> $^{2}$) we find and In Sec. 4 we defined ghost surfaces in terms of an action-gradient flow in GLYPH<18> , but defined QFMin surfaces in terms of GLYPH<2> . Here we further unify the formulation by defin- ing the action-gradient flow also in terms of GLYPH<2> , where T is the timelike label for GLYPH<2> ghost orbits [so dT = T $^{0}$( GLYPH<2>$_{0}$ ) d GLYPH<2>$_{0}$ ] and the inverse function # ( GLYPH<2> ; GLYPH<16> ) is defined by solving Eq. (8) for GLYPH<18> as a function of GLYPH<2> . From Eq. (10) we also have, for corrected QFMin or- bits, where N GLYPH<2> m ; m $^{0}$; n ; n 0 GLYPH<17> m $^{0}$( m + m $^{0}$) $^{h}$m + m 0 GLYPH<0> m $^{0}$GLYPH<14>$_{(}$$_{m}$$_{+}$$_{m}$ $^{0}$) p ; ( n + n $^{0}$) q$^{i}$ (the sign of the GLYPH<14>$_{(}$$_{m}$$_{+}$$_{m}$ $^{0}$) p ; ( n + n $^{0}$) q term in this expression being the opposite of that for GLYPH<18> ghost orbits). Eliminating GLYPH<14> S =GLYPH<14> GLYPH<2> between Eq. (18) and Eq. (19) we find the new reconciliation condition We have reviewed the motivation and formulation of a recently published [4] unification of ghost and quadratic- flux-minimizing surfaces, and have presented a new for- mulation that appears more natural and more useful in that it could be used to form a straight pseudo-field-line coor- dinate system. where, in the second equality, we have defined GLYPH<2>$_{0}$ so that T $^{0}$( GLYPH<2>$_{0}$ ) GLYPH<23> ( GLYPH<2>$_{0}$ ) GLYPH<17> GLYPH<0> 1, which satisfies the solvability condi- tion that $_{R}$2 GLYPH<25> 0 d GLYPH<2>$_{0}$ D GLYPH<2> = D GLYPH<2>$_{0}$ = 2 GLYPH<25> without the need to intro- duce a factor like the GLYPH<22> ( GLYPH<16> ) required in Sec. 5. Furthermore, Eq. (20) is satisfied by choosing the GLYPH<16> - dependence of the pseudo-orbit labeling so that More work remains to be done to develop a numeri- cal algorithm for constructing such ghost / QFMin surfaces at arbitrary order of perturbation, and to show they pre- serve the desirable properties of our previous formulation of ghost surfaces. The relation to heat transport in chaotic fields also needs to be investigated more deeply. conjugating the GLYPH<18> dynamics to straight-field-line dynamics (cf. Ref 6). Thus the GLYPH<2> ghost surface approach appears to be much superior to GLYPH<18> ghost surface approach, but needs to be tested in practice. As a first test we adapt the perturbative calculation in Sec. 6 to GLYPH<2> ghost surfaces. First, expanding # ( GLYPH<2> ; GLYPH<16> ) = GLYPH<2> + $^{P}$m ; $_{n}$GLYPH<18>$_{m}$$_{;}$$_{n}$ exp( im GLYPH<2> GLYPH<0> in GLYPH<16> ) and using Eq. (21), we recover Eq. (16). This ansatz is to be inserted in Eq. (19), written in the form From Eq. (14), dL$_{˙}$ GLYPH<18> = d GLYPH<16> = GLYPH<0> $^{P}$m ; $_{n}$(m GLYPH<19> - GLYPH<0> n )$^{2}$GLYPH<18>$_{m}$$_{;}$$_{n}$ exp[ im GLYPH<2>$_{0}$ + i ( m GLYPH<19> - GLYPH<0> n ) GLYPH<16> ], L$_{GLYPH<18>}$ = GLYPH<0> GLYPH<15> $^{P}$m ; $_{n}$imV$_{m}$$_{;}$$_{n}$ exp[ im GLYPH<18> ( GLYPH<16> j GLYPH<2>$_{0}$ ) GLYPH<0> in GLYPH<16> ], and [1] S. R. Hudson and J. Breslau, Phys. Rev. Lett. 100 , 095001 (2008). [2] S. R. Hudson and R. L. Dewar, J. Plasma Phys. 56 , 361 (1996). [3] R. L. Dewar, S. R. Hudson, and P. F. Price, Phys. Lett. A 194 , 49 (1994). [4] S. R. Hudson and R. L. Dewar, Phys. Lett. A 373 , 4409 (2009). [5] C. Gol´e, Symplectic Twist Maps : Global Variational Tech- niques (World Scientific, Singapore, 2001). [6] R. L. Dewar and J. D. Meiss, Physica D 57 , 476 (1992).