- Email: [email protected]
On Euclidean spinors and Wick rotations
5 December 1996
PHYSICS
LETTERS B
Physics Letters B 389 (19%) 29-36
On Euclidean spinors and Wick rotations * Peter van Nieuwenhuizen
l,
Andrew Waldron 2
Institute for Theoretical Physics, State Universi@ of New York at Stony Brook, Stony Brook, NY I 1794-3840, USA
Received 5 September 1996 Editor: M. Die
Abstract We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosom. The result is a recipe to construct a supersymmettic Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a fivedimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder-Schrader approach in which spinors are “doubled” but the action is not hermitian. However, for Dirac spinors our work provides a link to the work of Schwinger and Zumino in which hermiticity is maintained but spinors are not doubled. Our work differs from recent work by Mehta since we introduce no external metric and transform only the basic fields. PACS: 11.10.-z; 11.3O.W; 11.90.+t
1. Introduction Euclidean quantum field theory for bosons is used in particle physics to justify certain manipulations in Minkowski quantum field theory, such as the claim that for t -P foe only the vacuum contributes to the transition amplitude [ 11. (Sometimes a term ie+2 is added to the action by hand to obtain the same result in Minkowski spacetime, but this approach is somewhat ad hoc [ 21) . Furthermore Euclidean field theory is crucial for the physics of instantons. If one couples fermions to instantons, one needs Euclidean field theory for spinors. In fact, an N = 2 supersymmetric field theory has been constructed in Euclidean space [ 31. *This research was supported in part by NSF grant no. PHY9309888. 1Email: [email protected]. * E-mail: wnllyOinsti.physics.sunysb.edu.
Also finite temperature field theory and lattice gauge theory [4] use Euclidean field theory. In Fujikawa’s approach [ 51 to anomalies in quantum field theories one must regulate the Jacobian with a Gaussian integral whose part quadratic in momenta should be converging in all directions. To find the subleading terms (for example, terms with A,ys or iA,ys) one would like to know the continuation of the field theory to Euclidean space. Some mathematical applications of recent work by Seiberg and Witten are also based on supersymmetric theories continued to Euclidean space [ 61. In this article we present a new interpretation of the Wick rotation to Euclidean field theory. To avoid misunderstanding, we stress that the object of our study is not the Wick rotation of the momentum variable ko in a Feynman amplitude; rather, we are interested in the Wick rotation of the underlying field theory.
0370-2693/96/$12.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved. PII SO370-2693(96)01251-8
P. van Nieuwenhuizen, A. Waldron / Physics Letters B 389 (1996) 29-36
30
For spinors a quantum field theory in Euclidean space whose Green’s functions coincide with those of Minkowski spacetime after analytic continuation in the time coordinate, was constructed by Osterwalder and Schrader (OS) in 1973 [ 71. Their work was based on canonical quantization rather than path integrals and is, as a consequence, rather complicated. For example (i) a spinor field in four dimensional Euclidean space is expanded in creation and annihilation operators depending on Euclidean four-momenta kz, rather than three momenta k as in the Minkowski case, (ii) fields do not (and cannot) obey field equations, (iii) instead of the familiar spinors U: (k) and ~2( -k) (where I = fandcu= 1,. . . ,4 is the spinor index) of Minkowski spacetime, one now has spinors Uf (kz) , V,”( -kF), Xt(kk) and Wit-k:) whereR=1,...,4. However, if one rephrases their work in the language of path integrals, then the only new aspect is that one should “double” the number of spinor fields in Euclidean space. For example, for a Dirac spinor the action in Minkowski spacetime and in Euclidean space, respectively, reads &, = -$+i~O(~P”a, + m)$; /_L,v=o ,...,
{yIL,yV} = 2?j@‘;
3,
where ,yi is not related to $E by complex or hermitian conjugation, We have written the word “double” in quotation marks because the number of Grassmann integration variables is the same: 9 and $+ in one case, and # and x+ in the other3 . The only property that has changed is hermiticity4. The Euclidean action of OS is not hermitian. There are other approaches to Euclidean spinors, due to Schwinger [ 81 and Zumino [ 31; Fubini, Hanson and Jackiw [ 91; and more recently Mehta [ 101. Nicolai [ 111 extends the work of OS to Majorana 3Notice that in the path integral Xt and 9 are treated as independent variables while in the canonical approach they contain the same (doubled) creation and annihilation operators. This is just like I$ and @t in Minkowski space. 4 For anti-commuting variables we define (xt@)t = Jltx. Our convention for the metric signature in Minkowski spacetime is vau = diag(-,
+, +, +),,.
spinors 5 and defines the following action in Euclidean space
where @Eno longer satisfies a reality condition and C is the charge conjugation matrix which can equally well be defined in Minkowski and Euclidean space (it satisfies C T = -C and Cy,C-’ = -r,‘). In Minkowski spacetime one imposes the condition $+irO = I+Q~Cwhich yields a hermitian action, but again LE( Maj) is not hermitian. Zumino required that the Euclidean action be hermitian (which is not necessary on physical grounds) and since no Euclidean Majorana spinors exist (by which we mean spinors in Euclidean space satisfying a reality condition), he considered an N = 2 supersymmetric model with Dirac spinors, whose kinetic term for the spinor fields coincides with the original hermitian action proposed by Schwinger. Fubini et al. use “radial quantization”, in which the radius in Euclidean space plays the rale of the time coordinate. (In string theory one usually continues the world-sheet time t to --in, and makes then a conformal map to the “z-plane”. This is not the radial quantization of [ 9] ). As far as we know, Mehta was the first to attempt to write down a continuous one-parameter family of actions Lo such that Ce=o is the Minkowski action and f&,/2 is Schwinger’s Euclidean action (although he seems to be unaware of Schwinger’s work). In his approach, the dependence of the Dirac matrices on the parameter 8 is not a unitary rotation, indeed it is even discontinuous at the midpoint 8 = 7r/4. An “interpolating metric” (g’),,, is also introduced by hand such that the Euclidean action is hermitian, SO(4) invariant and without fermion doubling. For an application of Euclidean field theory to stochastic processes see [ 131, for a superfield formulation of Euclidean supersymmetry see [ 141 and for an approach that uses zero modes in Euclidean space to compute anomalies, see [ 151. We shall later comment on the relation between some of the above approaches, but first we shall introduce ours.
5 For a similar approach
in two dimensions
see
[ 121.
P.uanNieuwenhuizen,
A. Waldron/Physics
2. A new Wick rotation for Dirac spinors
In our construction we transform only the basic fields. Hence no extraneous metrics &, are introduced. The transformation induced on the Dirac matrices is unitary. The basic observation from which we start is that for a complex vector field W, one continues the combination W;IWp = - Wo WO + WC?Wi (which appears in the action) from Minkowski to Euclidean space by giving both W, and W; the same Wick rotation phase ei8. If 6 t + eeier with r the Euclidean time coordinate at 0 = 7r/2, Wo(t,x)
+ ei0W~(7,x),
(3)
Wi(t,x)
+ PW~**T,X),
(4)
At 0 = 7r/2, one then obtains Wf* WF+ Wr* WF where we call wi=r’* = WF the fourth component of the Euclidean field 7. We view this transformation for vector fields as a matrix which is diagonal with entries (e’“, 1, 1, 1). This suggests to define a Wick rotation for a general field as in the theory of induced representations: by transforming its coordinates (the orbital part) and its indices (the spin part). It is just an accident that for vector fields the spin matrix is diagonal, but in general the spin matrix may be more complicated. These observations suggest that one should consider the following Wick rotation for a Dirac spinor rCl(t,x) 4 S(@IcIs(r, x) 9
(5)
$+(tJ)
(6)
---$ti;(rJ)s(&
where 1,&./2 5 $n is the Euclidean Dirac spinor. S( 0) is a 4 x 4 matrix acting on spinor indices. Since the matrix S( 8) is diagonal in the vector case, there is at this point an ambiguity whether we should use S( 0) or sT( 8) in (6). At first sight one might expect that the correct relation should invoIve ST (8) in order that 6The arrow denotes a substitution to be made in the action; we do not employ an equal sign since in equations like t 4 emi@ we consider both t and r to be real. 7 A formal way to achieve this is to introduce “hyperbolic complex” numbers of the form a + eb where n and b are real and e satisfies e* = -e, but e* = +l [ 161. If one then replaces the transformation law WO -+ e"W4(0) by W,,-+ eieoW4(0) = (cost9+iesin@)W4(8),one finds W;Wo -) ezdW,l(0)W4(O) so that at t9 = 7r/2 one obtains -WI W4.
Leners B 389 (19%)
31
29-36
the group property holds. However, since the group is abelian (there is only one parameter 0)) also S( 0) is possible. In fact, the correct relation involves S( 8) as in (6) and ST (8) is inconsistent, as we shall show. In order that at 8 = 7r/2 the Euclidean action contains the operator
z rl(e)yks(e)
y4(e) 5 s-1(e)y4s(e)
= yk, = y4c0sB+y5sint9,
(7) (8)
satisfy the Euclidean Clifford algebra. (The i in 8, = id, converts y” into y” = iy”.) In order that the action at 6 = r/2 has an SO(4) rather than SO( 3,1) symmetry we require that M commutes with the SO(4) generators [ $, yi] . Hence h4 is proportional to a linear combination of the unit matrix and gE. It is natural to assume that the Wick rotation matrix S(0) should depend only on y” and y5 because the Wick rotation does not act on the space sector. The general case M = a1 + @r’, does not lead to an hermitian action and we continue with M N 4. The solution for the spin matrix S( 0) is given by s(e) = $J@/*; y5 = (ys)+, y4 = (y4>+,
(y5)2 = 1;
(y4)2 = 1.
(9)
Then
(10)
where = r/2),
(11)
7: = s-* (e = rr/2)y4s(e = 42).
(12)
7; = s-t (e = ~/2)ykqe
It is now clear that the choice 9+ --f &ST (where S 3 S(0 = a/2) = eY4gr/4) is not possible since defining STy4 = (a1 + &)S-’ and using gn = S-‘9s leads to SST = ay4 + /3ysy4 which is not consistent as in a general representation the right hand side is not symmetric.
8Our Euclidean signature is (+ + ++) so Euclidean indices may be raised and lowered with impunity, i.e. x4 = x4. We define y4 z ifl and ys = ~‘yays#, similarly 2 = ykga?n$.
P. van Nieuwenhuizen,A. Waldron/PhysicsLettersB 389 (1996) 29-36
32 Clearly, S(B)y4
S(B)
= y”S-’
is unitary for all real 0, and satisfies (f3).
Hence y: is hermitian. We note that S( 0) describes a rotation in the “plane” spanned by y4 and 9. In particular, at B = r/2, &=yk,
y;=ys.
(13)
Furthermore ?a = -7”. It clearly satisfies (r;)+ = Y5 E7
(r;)* = 1,
{Y&$1 = 0
3. Dirac spinors
(14)
and quantities such as ( 1 + y5) /2 appearing in an action transform into ( 1 + y5)/2 = ( 1 - y4)/2. Our Euclidean action can then finally be written as CE=(6~(7,X)YSE(~a,+m)~(7,x>.
induces a compensating Lorentz transformation in the 4-5 plane. Our Wick rotation matrix S(8) is just the spinor representation of this complex finite Lorentz boost. Further interesting questions are raised if we consider our Wick rotation in curved space.
(15)
This is the hermitian action Schwinger wrote down in 1959, but for eight component spinors. He considered real four component spinors in Minkowski spacetime but in order to still be able to impose a reality condition on Euclidean spinors he found it necessary to double the number of spinor components in Euclidean space. He introduced a matrix exp( &I$?riy”yoy5 @ g3) whose role is to interchange some Dirac matrices such that one ends up with real symmetric Dirac matrices in Euclidean space. We have given a derivation which starts from transformation rules of the fields, which is clearly more fundamental, but note that our matrix qe = r/2) is the same as Schwinger’s matrix. We keep the number of components of spinors equal to four, but since a Dirac spinor is equivalent to an eight component real spinor the Euclidean action we obtain is equivalent to Schwinger’s action. Also in Zumino’s work, the YE is not explicit since he considered massless spinors and worked with Dirac matrices 7: satisfying {yg, ys} = 2P”. From our perspective $ = ;$ux rather than < = $. The procedure we have followed to obtain Euclidean spinors from their Minkowski counterparts boils down to the analytic continuation t d -ir and a simultaneous rotation of the spinor indices. In this respect we differ from the work of OS who continue the Minkowski theory in time only and then construct a corresponding canonical Euclidean field theory. For vector fields one rotates the indices exactly as if 7 = it were part of a general coordinate transformation in the tr plane (t’ = it yields W,$(t’) c = -iWo( t) ). Requiring that the = $Wo(t’) W4(t’) flat space vielbein field e,,“’ = 8,” does not change,
Zumino [ 31 constructed a supersymmetric Euclidean field theory in 1977 which follows Schwinger’s approach. The aim was to incorporate instantons into supersymmetric theories. The N = 2 supersymmetric action in Euclidean space obtained by him contains the fields A (scalar), B (pseudoscalar), V’ (real vector) and r// (Dirac spinor) just as for the Minkowski spacetime action. However the actions differ in crucial parts. In Minkowski spacetime the N = 2 supersymmetric Yang Mills action reads CM = -iFEY - ;(DpA)’ - ig2(A
- ;(DllB)*
x B)2 - $+y4 .@@
-g$+y4.(iA+B?)
x$,
(16)
where D,A = >,A i- gV’ x A, idem D,B and D,&, and F& = -2F$ + F;. The Minkowski N = 2 supersymmetry transformations are S$ =$AE + iy’jbBe - ~y*y’F,,r - g(A x B)y5e
(171
(18)
(19) (20) (21)
In Euclidean space the following hermitian and supersymmetric action was constructed 9 9To match the notations in (22)-(27) with those of [31, replace A (the reason for this permutation will soon become clear), i$< + yp, -gE - 9 ($ - igyp) and the Euclidean supersymmetry parameter EE - -k.
,+ 4 8, BE -
P. van Nieuwenhuizen, A. Wakfron/ Physics Letters B 389 (1996) 29-36
+ ;(D,B~)~ + ig2(A~x BE)~+ $iy;IE .gEh i-i&f& (T&E+ BE) X ilf~,
J&=-7
VE24i
$F;~
-
;(D,A~)~
1
(22)
which is invariant under the following transformation rules
(24)
-I- igE+$E(AE X BE),
33
is clearly compatible with the Wick rotation rules of $ and $+ since the N = 2 supersymmetry parameter E itself is a Dirac spinor. Further we note that our Wick rotation is consistent with the hermiticity assignments of the continued Euclidean fields. This is a non-trivial statement, consider the supersymmetry transformation rule Sljl = ME for some matrix M. To obtain the rule for SI& one can proceed in two ways. Either one first constructs S$E and then takes the hermitian conjugate, or one first takes the hermitian conjugate of Sfr and then continues. In the latter case one should use (6). Hence S@ = ME * S&& = ( M)ESEE * SI+!Q= S-’ ( M)J$EE
MAE=
-E~Y&+E
ABE
=
t E&E
sv;
=
i(&@E
-
(25)
&$EE,
* SI,!J~= ENS-’ (ME) +S,
t -I- &EE,
(26)
+ &&~eE)
(27)
’
Note that the transformations RAE, ABE and SVj are all hermitian, while && and &,@are related by hermitian conjugation. We claim that this result follows unambiguously from our Wick rotation. To see this we make the following transformations in the Minkowski action (we give only the finite transformations but one may reinstate the Wick rotation angle 0 as outlined above to obtain a continuous Wick rotation) $ ---),r47w4~E; V’ = (Vo, V), A 4 AE;
$t
_+
&Y4?%4
--t (iv,“, VE)P;
B +iBE.
;
d4x --$ -id4x; (28)
The i in the line for Vo is the i from Eq. (4), but the i in the line for the pseudoscalar B needs a short expla-
nation. If a pseudoscalar is represented in Minkowski spacetime in the form PpcraP& Q$2aP&&~~, then under the Wick rotation t 4 47, the field B goes over into iBE since only fields, not constants such as eC”PO,are transformed in our approach. It is straightforward to check that the substitution of the transformations (28) into the Minkowski action ( 16) yield the Euclidean action (22). Under the same Wick rotation rules, the Minkowski supersymmetry transformations go over to those of the Euclidean theory provided we at the same time also replace E ---, .?r4’“f4EE and e+ ---) E~$Y~$“/~.This rule
S$ = ME j j
S&S
(29)
&,h+ = E+M+ = &(
M+),
* S& = E$‘(M’)ES-’
(30)
(where S = e7475v/4 = S( 0 = 7r/2) ) . Since the matrix M may depend on fields and/or derivatives its continuation ME is in general not equal to M. Consistency requires S-‘(ME)+S=
S(M+)&‘.
(31)
One may check that the matrix M of the Minkowski supersymmetry transformation rule Se in ( 17) satisfies this consistency condition. It is interesting to note that just as non-compact SO( 3,1) Minkowski spacetime Lorentz transformations become compact SO(4) rotations in Euclidean space, in contradistinction the compact axial V( 1) transformations of the Minkowski theory become non-compact scale transformations in the Euclidean case [3,10]. The requirement of hermiticity of the Euclidean action is less justified than the property which Osterwalder and Schrader (and Nicolai) imposed, namely that the Greens functions in Minkowski and Euclidean space are each other’s analytic continuation. We have shown that our approach happens to satisfy both requirements simultaneously for Dirac spinors. The way we have achieved this is by not only continuing analytically the time coordinate as in OS, but also rotating spinor indices (with the matrix S) . Finally let
P. uan Nieuwenhuizen, A. Waldron / Physics Letters B 389 (19%) 29-36
34
US observe that Zumino’s Euclidean pseudoscalar kinetic action +1/2( 8pB)2 is not damped in the Euclidean path integral (recall that expi Jd4xLM + exp / d4x&). In this light one may prefer to trade the requirement of hermiticity (of the action and supersymmetry transformations) for damping by introducing a field Bf, = iBE.
4. Majorana
and Weyl spinors
Within a path integral approach, Nicolai [ 1 l] obtained an action for Majorana spinors depending only on a single spinor ti (and not $I+) without any reality condition, &@‘faJ)
1 = -&‘E
T
c(Y;+
+ m)tlrE,
(32)
where C is the Minkowski conjugation matrix and his Dn-ac matrices YE = (Y’ , y2, y3, y”) undergo no rotation. In this framework, he is still able to write down a supersymmetric action since, within an N = 1 supersymmetric model, the reality of en is not needed to verify invariance under supersymmetry transformations. Further, the various Fierz identities required are unchanged since the charge conjugation matrix C is still that of the Minkowski theory. In our approach to Majorana spinors we begin with the usual Minkowski action &(Maj)
= -$$TC
+m>$.
= -$@STCSS-1(yP6’P+rn)S$E
= -$tffETcdY
%$+ m)+E
(34)
CE = STCS also satisfies the relations CE = -Cz and CnYgC<’ = -YEPT. Formally this action is the same as that of Nicolai except that Nicolai performs no rotation on spinor indices (and therefore his charge conjugation matrix is the original Minkowski charge conjugation matrix). Neither action is hermitian and in neither case are the classical fields $ and rJ+ related by a reality condition. In an expanded version we shall work through the example of the N = 1 Wess Zumino model. and the matrix
of?, &L.R
‘kL’L,R= +;
1 &y5 X;,R =Xt-’
2’
/&ys 7
= f+L.R
;
= &X+L,R .’
(35) (36)
where @ and x+ are unconstrained Dirac spinors. Transforming (/I ---f S$n and xt + x$ (as usual S = eY4Ys.rr/4)in accordance with our Wick rotation rules, we obtain the transformation rules for Weyl spinors
(33)
Dropping the reality constraint, we can now unambiguously perform our Wick rotation on r/~ as given above. We find &(Maj)
In Minkowski spacetime one may always rewrite a Weyl spinor as a Majorana spinor. However, we found it necessary to drop the reality condition on Majorana spinors when continuing to Euclidean space. The Berezin integration of the path integral is not sensitive to this modification and in this way we were able to reproduce continued Greens functions. We now consider the Wick rotation for Weyl spinors. In Minkowski spacetime$t_=~(1+$)$and@~=$+~(1+y5> is the hermitian conjugate of (GIL.As suggested by the analysis for Majorana spinors, we should first drop this hermiticity requirement and then perform the Wick rotation as for Dirac spinors. Therefore we consider spinors $ and x+, unrelated by hermitian or complex conjugation conjugation but nevertheless eigenstates
where we used s-’
1 iY5 Fs=y
l&Y;
(and similarly for S and S-’ interchanged). Consider now the Minkowski action for a left-handed Weyl spinor (in which we have already relaxed hermiticity by replacing $+ ---f xt) &(Weyl,
I.,) = -x~Y”##L
= -X+Y4# Fe. ’ + y5 (39)
Under Wick rotation we obtain &(Weyl,L)
= X~Y~(“~@s
= -XEt#E#EL. (40)
P. van Nieuwenhuizen, A. Waldron /Physics
where in Euclidean space
As in the the Majorana case this action is not (and cannot be) hermitian. Further notice that the x[ * &, coupling of Minkowski space becomes xi H $r_ in Euclidean space [ lo]. In an expanded version we shall apply this approach to N = 1 super Yang Mills theory. 5. Conclusion
We have defined a continuous Wick rotation on Dirac spinors $ and their hermitian conjugates by means of a Lorentz transformation S(0) in the Euclidean-Minkowski tr plane. The resulting Euclidean Greens functions are not only obtained from their Minkowski counterparts by continuation in the time variable, but also by a rotation of spinor indices. In this respect our work agrees with work of Schwinger, who, however, did not implement this idea on the basic fields. For Majorana and Weyl spinors however, one is forced to drop the reality/hermiticity condition if one wants to perform the Wick rotation and hence our formalism, which still uses the Lorentz rotation S, now resembles the Osterwalder-Schrader approach. Our final rule for the Wick rotation on spinors is $Vr,x)
4 S(@$e(r,x),
(41)
ICl+(tJ) h &rJ)s(e)
(42)
with S(6) = ,Y4YSs/*.Using s(fl)y4 = y4S-’ (d), the Dirac Minkowski action goes over into
j
-‘&4(‘&
3 k + ?@4
+ m)@E
(43)
and since we found that r4 = -$, the action is clearly hermitian and SO(4) invariant. Thus the troublesome 7” is not omitted in Euclidean space as one might expect, but rather reinterpreted as gE in Euclidean space [lo]. With the understanding of the Wick rotation provided by our formalism one can explicitly and continuously follow how vertices and propagators in in-
Letters B 389 (1996) 29-36
35
teracting field theories change as one rotates from the Minkowski to the Euclidean case. This has practical advantages, for example it explains which terms with factors of 9 and axial vector fields in regulators for anomalies acquire factors of i. Another useful application is supersymmetry; our rules automatically generate N = 1 and N = 2 theories in Euclidean space whenever they exist in Minkowski space. Also, another obvious open question is how to generalize our Wick rotation to odd dimensions in which there is no “g”. One possibility is to double the Dirac matrices, which suggests that the cases in which one is able to operate with undoubled spinor degrees of freedom in Euclidean space are the exception rather than the norm [ 181. In this article we have discussed only the path integral aspects of the Wick rotation. The OsterwalderSchrader approach is entirely based on a canonical approach, and our Wick rotation can also be formulated in a canonical way, but this will be published elsewhere [ 181. There we also discuss the OsterwalderSchrader reflection positivity axiom which is the main ingredient for establishing the existence of a positive semi-definite self-adjoint Hamiltonian lo . The last comment we wish to make is perhaps the most interesting but certainly the least understood and speculative. Both in our work and also in that of Osterwalder-Schrader, one keeps noticing five-dimensional aspects tt . For example, in the OS canonical approach, the expansion into creation and annihilation operators has a normalisation factor (k* + m*>-l/* in Euclidean space, as opposed to (k’+m*)-I/* m ’ Minkowski spacetime. The former is clearly the usual normalization of a five-dimensional theory at t = 0, with four space momenta k;. Further Schwinger introduces new fields B(x) for the original fields A(x) which satisfy the commutation loIn more detail, this axiom requites that there exists an antilinear mapping 0 which transforms an arbitrary function F of the fields at positive times into a function OF of fields at negative times such that (OFF) is semi-positive. If such a mapping exists one can define a Hiibert space with positive norm and a positive transfer matrix. The choice of time axis with respect to which time is defined is arbitrary and breaks the four-dimensional symmetry but the results do not depend on the axis chosen [ 191. I1 It may be interesting also to recall the work of [ 201 in which the four-dimensional Wess Zumino model is obtained by dimensional reduction “by Legendre transformation” from the on-shell theory five dimensions
36
P. oan Nieuwenhuizen. A. Waldron /Physics
relations [A(x),B(y)] = #(x - y), which again can be viewed as an equal time canonical commutation relation in five dimensions. We found a Wick rotation from Minkowski to Euclidean space which is a five-dimensional Lorentz rotation, but there are many more questions and ideas yet to be developed. Acknowledgements It is a pleasure to thank R. Schrader, P Breitenlohner, D. Maison, E. Seiler, C. Preitschopf and B. Zumino for discussions.
References [ 11 C. Itzykson and J. Zuber, Quantum Field Theory (McGraw
[2]
[3] [4]
[5]
[6] [7]
Hill, 1980); E. Abers and B. Lee, Phys. Rep. 9 ( 1973) 1. L. Ryder, Quantum Theory of Fields (Cambridge University Press, 1985); P Ramond, Field Theory: a modem primer (Addison Wesley, 1990). B. Zumino, Phys. Lett. B 69 (1977) 369. For an application of the Osterwalder-Schrader formalism to lattice fermions see K. Osterwalder and E. Seiler, Ann. Phys. 110 (1978) 440; E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics (SpringerVerlag, Berlin, 1982). K. Fujikawa, Phys. Rev. L&t. 42 ( 1979) 1195; Phys. Rev. D 21 (1980) 2848; D 22 (1980) 1499; Phys. Rev. Lett. 44 (1989) 1733,65. N. Seiberg and E. Witten, Nucl. Phys. B 426 (1994) 19; B 430 ( 1994) 485 (E); Nucl. Phys. B 431 ( 1994) 484. K. Osterwalder and R. Schrader, Phys. Rev. L&t. 29 (1972) 1423; Helv. Phys. Acta 46 (1973) 277; CMP 31 (1973) 83; 42 (1975) 281; K. Osterwalder, Euclidean Greens fuctions and Wightman distributions; in: Constructive Field Theory - Erice lectures 1973, eds. G. Velo and A. Wightman, (Springer-Verlag, Berlin, 1973); for a more recent discussion see K. Osterwalder, in: Advances in Dynamical Systems and Quantum Physics, Capri Conference (World Scientific, 1993).
Letters B 389 (I 996) 29-36
J. Schwinger, Phys. Rev. 1IS (1959) 721; 117 (1960) 1407. The first paper contains the action for a Dirac spinor with a 9 matrix, see page 727 and the functional W on page 728; see also Proc. Nat. AC. of SC. 44 (1958) 956, where different analytic continuations of the time variable for timeordered or anti-time-ordered Minkowski Greens functions are considered. ]91 S. Fubini, A. Hanson and R. Jackiw, Phys. Rev. D 7 (1973) 1732; R. Jackiw, in: Erice 1973, Proceedings, Laws Of Hadronic Matter, New York 1975. In this work, fields are canonically quantized on equal proper time hyperboloids and subsequently continued to imaginary time so that the quantization becomes radial. [lOI M. Mehta, Phys. Rev. Leu. 65 (1990) 1983; 66 (1991) 522; Mod. Phys. L&t. A 30 ( 1991) 28 11; Phys. Lett. B 274 (1991) 53. IllI H. Nicolai, Nucl. Phys. B 140 (1978) 294; B 156 ( 1979) 157, 177; Phys. l&t. B 89 ( 1980) 341; for earlier work on Euclidean Majorana spinors see also J. Friihlich and K. Osterwalder, Helv. Phys. Acta, 47 ( 1974) 781. LIZI D. Borthwick, J. Math. Phys. 34 ( 1993) 2691. This paper applies Nicolai’s Majorana version of the OS formalism to two dimensions; for results for the Yukawa model in two dimensions see E. Seiler, Commun. Math. Phys. 42 (1975) 163. [I31 J. Kupsch and W.D. Thacker, Fortschr. Phys. 38 ( 1990) 35; Z. Haba and J. Kupsch, Fortschr. Phys. 43 ( 1995) 41; I. Kupsch, J. Geom. and Phys. 11 (1993) 507; Ind. J. Phys. A 79 (1996) 293; Rev. Math. Phys. 2 (1990) 457. [I41 J. Lukierski and A. Novicki, J. Math. Phys. 25 ( 1984) 2545. [I51 H. Banerjee, Z. Phys. C 62 ( 1994) 511, Resolution of the strong CP and U( I ) problems, hep-th/9508065 [ 161 G. Gibbons, M. Green and M. Perry, Phys. Lett. B 370 (1996) 37. [I71 R. Streeter and A. Wightman, PCT spin, statistics and all that (Benjamin, New York, 1964). [I81 A. Waldron, to appear. [I91 I. Montvay and G. Miinster, Quantum Fields on a Lattice (Cambridge University Press, 1994). 1201 M. Sohnius, K. Stelle and P. West. Nucl. Phys. B 173 ( 1980) 127; I. Martin-Hemandez and J.G. Taylor, Phys. Lett. B 18.5 (1987) 99.
PHYSICS
LETTERS B
Physics Letters B 389 (19%) 29-36
On Euclidean spinors and Wick rotations * Peter van Nieuwenhuizen
l,
Andrew Waldron 2
Institute for Theoretical Physics, State Universi@ of New York at Stony Brook, Stony Brook, NY I 1794-3840, USA
Received 5 September 1996 Editor: M. Die
Abstract We propose a continuous Wick rotation for Dirac, Majorana and Weyl spinors from Minkowski spacetime to Euclidean space which treats fermions on the same footing as bosom. The result is a recipe to construct a supersymmettic Euclidean theory from any supersymmetric Minkowski theory. This Wick rotation is identified as a complex Lorentz boost in a fivedimensional space and acts uniformly on bosons and fermions. For Majorana and Weyl spinors our approach is reminiscent of the traditional Osterwalder-Schrader approach in which spinors are “doubled” but the action is not hermitian. However, for Dirac spinors our work provides a link to the work of Schwinger and Zumino in which hermiticity is maintained but spinors are not doubled. Our work differs from recent work by Mehta since we introduce no external metric and transform only the basic fields. PACS: 11.10.-z; 11.3O.W; 11.90.+t
1. Introduction Euclidean quantum field theory for bosons is used in particle physics to justify certain manipulations in Minkowski quantum field theory, such as the claim that for t -P foe only the vacuum contributes to the transition amplitude [ 11. (Sometimes a term ie+2 is added to the action by hand to obtain the same result in Minkowski spacetime, but this approach is somewhat ad hoc [ 21) . Furthermore Euclidean field theory is crucial for the physics of instantons. If one couples fermions to instantons, one needs Euclidean field theory for spinors. In fact, an N = 2 supersymmetric field theory has been constructed in Euclidean space [ 31. *This research was supported in part by NSF grant no. PHY9309888. 1Email: [email protected]. * E-mail: wnllyOinsti.physics.sunysb.edu.
Also finite temperature field theory and lattice gauge theory [4] use Euclidean field theory. In Fujikawa’s approach [ 51 to anomalies in quantum field theories one must regulate the Jacobian with a Gaussian integral whose part quadratic in momenta should be converging in all directions. To find the subleading terms (for example, terms with A,ys or iA,ys) one would like to know the continuation of the field theory to Euclidean space. Some mathematical applications of recent work by Seiberg and Witten are also based on supersymmetric theories continued to Euclidean space [ 61. In this article we present a new interpretation of the Wick rotation to Euclidean field theory. To avoid misunderstanding, we stress that the object of our study is not the Wick rotation of the momentum variable ko in a Feynman amplitude; rather, we are interested in the Wick rotation of the underlying field theory.
0370-2693/96/$12.00 Copyright @ 1996 Elsevier Science B.V. All rights reserved. PII SO370-2693(96)01251-8
P. van Nieuwenhuizen, A. Waldron / Physics Letters B 389 (1996) 29-36
30
For spinors a quantum field theory in Euclidean space whose Green’s functions coincide with those of Minkowski spacetime after analytic continuation in the time coordinate, was constructed by Osterwalder and Schrader (OS) in 1973 [ 71. Their work was based on canonical quantization rather than path integrals and is, as a consequence, rather complicated. For example (i) a spinor field in four dimensional Euclidean space is expanded in creation and annihilation operators depending on Euclidean four-momenta kz, rather than three momenta k as in the Minkowski case, (ii) fields do not (and cannot) obey field equations, (iii) instead of the familiar spinors U: (k) and ~2( -k) (where I = fandcu= 1,. . . ,4 is the spinor index) of Minkowski spacetime, one now has spinors Uf (kz) , V,”( -kF), Xt(kk) and Wit-k:) whereR=1,...,4. However, if one rephrases their work in the language of path integrals, then the only new aspect is that one should “double” the number of spinor fields in Euclidean space. For example, for a Dirac spinor the action in Minkowski spacetime and in Euclidean space, respectively, reads &, = -$+i~O(~P”a, + m)$; /_L,v=o ,...,
{yIL,yV} = 2?j@‘;
3,
where ,yi is not related to $E by complex or hermitian conjugation, We have written the word “double” in quotation marks because the number of Grassmann integration variables is the same: 9 and $+ in one case, and # and x+ in the other3 . The only property that has changed is hermiticity4. The Euclidean action of OS is not hermitian. There are other approaches to Euclidean spinors, due to Schwinger [ 81 and Zumino [ 31; Fubini, Hanson and Jackiw [ 91; and more recently Mehta [ 101. Nicolai [ 111 extends the work of OS to Majorana 3Notice that in the path integral Xt and 9 are treated as independent variables while in the canonical approach they contain the same (doubled) creation and annihilation operators. This is just like I$ and @t in Minkowski space. 4 For anti-commuting variables we define (xt@)t = Jltx. Our convention for the metric signature in Minkowski spacetime is vau = diag(-,
+, +, +),,.
spinors 5 and defines the following action in Euclidean space
where @Eno longer satisfies a reality condition and C is the charge conjugation matrix which can equally well be defined in Minkowski and Euclidean space (it satisfies C T = -C and Cy,C-’ = -r,‘). In Minkowski spacetime one imposes the condition $+irO = I+Q~Cwhich yields a hermitian action, but again LE( Maj) is not hermitian. Zumino required that the Euclidean action be hermitian (which is not necessary on physical grounds) and since no Euclidean Majorana spinors exist (by which we mean spinors in Euclidean space satisfying a reality condition), he considered an N = 2 supersymmetric model with Dirac spinors, whose kinetic term for the spinor fields coincides with the original hermitian action proposed by Schwinger. Fubini et al. use “radial quantization”, in which the radius in Euclidean space plays the rale of the time coordinate. (In string theory one usually continues the world-sheet time t to --in, and makes then a conformal map to the “z-plane”. This is not the radial quantization of [ 9] ). As far as we know, Mehta was the first to attempt to write down a continuous one-parameter family of actions Lo such that Ce=o is the Minkowski action and f&,/2 is Schwinger’s Euclidean action (although he seems to be unaware of Schwinger’s work). In his approach, the dependence of the Dirac matrices on the parameter 8 is not a unitary rotation, indeed it is even discontinuous at the midpoint 8 = 7r/4. An “interpolating metric” (g’),,, is also introduced by hand such that the Euclidean action is hermitian, SO(4) invariant and without fermion doubling. For an application of Euclidean field theory to stochastic processes see [ 131, for a superfield formulation of Euclidean supersymmetry see [ 141 and for an approach that uses zero modes in Euclidean space to compute anomalies, see [ 151. We shall later comment on the relation between some of the above approaches, but first we shall introduce ours.
5 For a similar approach
in two dimensions
see
[ 121.
P.uanNieuwenhuizen,
A. Waldron/Physics
2. A new Wick rotation for Dirac spinors
In our construction we transform only the basic fields. Hence no extraneous metrics &, are introduced. The transformation induced on the Dirac matrices is unitary. The basic observation from which we start is that for a complex vector field W, one continues the combination W;IWp = - Wo WO + WC?Wi (which appears in the action) from Minkowski to Euclidean space by giving both W, and W; the same Wick rotation phase ei8. If 6 t + eeier with r the Euclidean time coordinate at 0 = 7r/2, Wo(t,x)
+ ei0W~(7,x),
(3)
Wi(t,x)
+ PW~**T,X),
(4)
At 0 = 7r/2, one then obtains Wf* WF+ Wr* WF where we call wi=r’* = WF the fourth component of the Euclidean field 7. We view this transformation for vector fields as a matrix which is diagonal with entries (e’“, 1, 1, 1). This suggests to define a Wick rotation for a general field as in the theory of induced representations: by transforming its coordinates (the orbital part) and its indices (the spin part). It is just an accident that for vector fields the spin matrix is diagonal, but in general the spin matrix may be more complicated. These observations suggest that one should consider the following Wick rotation for a Dirac spinor rCl(t,x) 4 S(@IcIs(r, x) 9
(5)
$+(tJ)
(6)
---$ti;(rJ)s(&
where 1,&./2 5 $n is the Euclidean Dirac spinor. S( 0) is a 4 x 4 matrix acting on spinor indices. Since the matrix S( 8) is diagonal in the vector case, there is at this point an ambiguity whether we should use S( 0) or sT( 8) in (6). At first sight one might expect that the correct relation should invoIve ST (8) in order that 6The arrow denotes a substitution to be made in the action; we do not employ an equal sign since in equations like t 4 emi@ we consider both t and r to be real. 7 A formal way to achieve this is to introduce “hyperbolic complex” numbers of the form a + eb where n and b are real and e satisfies e* = -e, but e* = +l [ 161. If one then replaces the transformation law WO -+ e"W4(0) by W,,-+ eieoW4(0) = (cost9+iesin@)W4(8),one finds W;Wo -) ezdW,l(0)W4(O) so that at t9 = 7r/2 one obtains -WI W4.
Leners B 389 (19%)
31
29-36
the group property holds. However, since the group is abelian (there is only one parameter 0)) also S( 0) is possible. In fact, the correct relation involves S( 8) as in (6) and ST (8) is inconsistent, as we shall show. In order that at 8 = 7r/2 the Euclidean action contains the operator
z rl(e)yks(e)
y4(e) 5 s-1(e)y4s(e)
= yk, = y4c0sB+y5sint9,
(7) (8)
satisfy the Euclidean Clifford algebra. (The i in 8, = id, converts y” into y” = iy”.) In order that the action at 6 = r/2 has an SO(4) rather than SO( 3,1) symmetry we require that M commutes with the SO(4) generators [ $, yi] . Hence h4 is proportional to a linear combination of the unit matrix and gE. It is natural to assume that the Wick rotation matrix S(0) should depend only on y” and y5 because the Wick rotation does not act on the space sector. The general case M = a1 + @r’, does not lead to an hermitian action and we continue with M N 4. The solution for the spin matrix S( 0) is given by s(e) = $J@/*; y5 = (ys)+, y4 = (y4>+,
(y5)2 = 1;
(y4)2 = 1.
(9)
Then
(10)
where = r/2),
(11)
7: = s-* (e = rr/2)y4s(e = 42).
(12)
7; = s-t (e = ~/2)ykqe
It is now clear that the choice 9+ --f &ST (where S 3 S(0 = a/2) = eY4gr/4) is not possible since defining STy4 = (a1 + &)S-’ and using gn = S-‘9s leads to SST = ay4 + /3ysy4 which is not consistent as in a general representation the right hand side is not symmetric.
8Our Euclidean signature is (+ + ++) so Euclidean indices may be raised and lowered with impunity, i.e. x4 = x4. We define y4 z ifl and ys = ~‘yays#, similarly 2 = ykga?n$.
P. van Nieuwenhuizen,A. Waldron/PhysicsLettersB 389 (1996) 29-36
32 Clearly, S(B)y4
S(B)
= y”S-’
is unitary for all real 0, and satisfies (f3).
Hence y: is hermitian. We note that S( 0) describes a rotation in the “plane” spanned by y4 and 9. In particular, at B = r/2, &=yk,
y;=ys.
(13)
Furthermore ?a = -7”. It clearly satisfies (r;)+ = Y5 E7
(r;)* = 1,
{Y&$1 = 0
3. Dirac spinors
(14)
and quantities such as ( 1 + y5) /2 appearing in an action transform into ( 1 + y5)/2 = ( 1 - y4)/2. Our Euclidean action can then finally be written as CE=(6~(7,X)YSE(~a,+m)~(7,x>.
induces a compensating Lorentz transformation in the 4-5 plane. Our Wick rotation matrix S(8) is just the spinor representation of this complex finite Lorentz boost. Further interesting questions are raised if we consider our Wick rotation in curved space.
(15)
This is the hermitian action Schwinger wrote down in 1959, but for eight component spinors. He considered real four component spinors in Minkowski spacetime but in order to still be able to impose a reality condition on Euclidean spinors he found it necessary to double the number of spinor components in Euclidean space. He introduced a matrix exp( &I$?riy”yoy5 @ g3) whose role is to interchange some Dirac matrices such that one ends up with real symmetric Dirac matrices in Euclidean space. We have given a derivation which starts from transformation rules of the fields, which is clearly more fundamental, but note that our matrix qe = r/2) is the same as Schwinger’s matrix. We keep the number of components of spinors equal to four, but since a Dirac spinor is equivalent to an eight component real spinor the Euclidean action we obtain is equivalent to Schwinger’s action. Also in Zumino’s work, the YE is not explicit since he considered massless spinors and worked with Dirac matrices 7: satisfying {yg, ys} = 2P”. From our perspective $ = ;$ux rather than < = $. The procedure we have followed to obtain Euclidean spinors from their Minkowski counterparts boils down to the analytic continuation t d -ir and a simultaneous rotation of the spinor indices. In this respect we differ from the work of OS who continue the Minkowski theory in time only and then construct a corresponding canonical Euclidean field theory. For vector fields one rotates the indices exactly as if 7 = it were part of a general coordinate transformation in the tr plane (t’ = it yields W,$(t’) c = -iWo( t) ). Requiring that the = $Wo(t’) W4(t’) flat space vielbein field e,,“’ = 8,” does not change,
Zumino [ 31 constructed a supersymmetric Euclidean field theory in 1977 which follows Schwinger’s approach. The aim was to incorporate instantons into supersymmetric theories. The N = 2 supersymmetric action in Euclidean space obtained by him contains the fields A (scalar), B (pseudoscalar), V’ (real vector) and r// (Dirac spinor) just as for the Minkowski spacetime action. However the actions differ in crucial parts. In Minkowski spacetime the N = 2 supersymmetric Yang Mills action reads CM = -iFEY - ;(DpA)’ - ig2(A
- ;(DllB)*
x B)2 - $+y4 .@@
-g$+y4.(iA+B?)
x$,
(16)
where D,A = >,A i- gV’ x A, idem D,B and D,&, and F& = -2F$ + F;. The Minkowski N = 2 supersymmetry transformations are S$ =$AE + iy’jbBe - ~y*y’F,,r - g(A x B)y5e
(171
(18)
(19) (20) (21)
In Euclidean space the following hermitian and supersymmetric action was constructed 9 9To match the notations in (22)-(27) with those of [31, replace A (the reason for this permutation will soon become clear), i$< + yp, -gE - 9 ($ - igyp) and the Euclidean supersymmetry parameter EE - -k.
,+ 4 8, BE -
P. van Nieuwenhuizen, A. Wakfron/ Physics Letters B 389 (1996) 29-36
+ ;(D,B~)~ + ig2(A~x BE)~+ $iy;IE .gEh i-i&f& (T&E+ BE) X ilf~,
J&=-7
VE24i
$F;~
-
;(D,A~)~
1
(22)
which is invariant under the following transformation rules
(24)
-I- igE+$E(AE X BE),
33
is clearly compatible with the Wick rotation rules of $ and $+ since the N = 2 supersymmetry parameter E itself is a Dirac spinor. Further we note that our Wick rotation is consistent with the hermiticity assignments of the continued Euclidean fields. This is a non-trivial statement, consider the supersymmetry transformation rule Sljl = ME for some matrix M. To obtain the rule for SI& one can proceed in two ways. Either one first constructs S$E and then takes the hermitian conjugate, or one first takes the hermitian conjugate of Sfr and then continues. In the latter case one should use (6). Hence S@ = ME * S&& = ( M)ESEE * SI+!Q= S-’ ( M)J$EE
MAE=
-E~Y&+E
ABE
=
t E&E
sv;
=
i(&@E
-
(25)
&$EE,
* SI,!J~= ENS-’ (ME) +S,
t -I- &EE,
(26)
+ &&~eE)
(27)
’
Note that the transformations RAE, ABE and SVj are all hermitian, while && and &,@are related by hermitian conjugation. We claim that this result follows unambiguously from our Wick rotation. To see this we make the following transformations in the Minkowski action (we give only the finite transformations but one may reinstate the Wick rotation angle 0 as outlined above to obtain a continuous Wick rotation) $ ---),r47w4~E; V’ = (Vo, V), A 4 AE;
$t
_+
&Y4?%4
--t (iv,“, VE)P;
B +iBE.
;
d4x --$ -id4x; (28)
The i in the line for Vo is the i from Eq. (4), but the i in the line for the pseudoscalar B needs a short expla-
nation. If a pseudoscalar is represented in Minkowski spacetime in the form PpcraP& Q$2aP&&~~, then under the Wick rotation t 4 47, the field B goes over into iBE since only fields, not constants such as eC”PO,are transformed in our approach. It is straightforward to check that the substitution of the transformations (28) into the Minkowski action ( 16) yield the Euclidean action (22). Under the same Wick rotation rules, the Minkowski supersymmetry transformations go over to those of the Euclidean theory provided we at the same time also replace E ---, .?r4’“f4EE and e+ ---) E~$Y~$“/~.This rule
S$ = ME j j
S&S
(29)
&,h+ = E+M+ = &(
M+),
* S& = E$‘(M’)ES-’
(30)
(where S = e7475v/4 = S( 0 = 7r/2) ) . Since the matrix M may depend on fields and/or derivatives its continuation ME is in general not equal to M. Consistency requires S-‘(ME)+S=
S(M+)&‘.
(31)
One may check that the matrix M of the Minkowski supersymmetry transformation rule Se in ( 17) satisfies this consistency condition. It is interesting to note that just as non-compact SO( 3,1) Minkowski spacetime Lorentz transformations become compact SO(4) rotations in Euclidean space, in contradistinction the compact axial V( 1) transformations of the Minkowski theory become non-compact scale transformations in the Euclidean case [3,10]. The requirement of hermiticity of the Euclidean action is less justified than the property which Osterwalder and Schrader (and Nicolai) imposed, namely that the Greens functions in Minkowski and Euclidean space are each other’s analytic continuation. We have shown that our approach happens to satisfy both requirements simultaneously for Dirac spinors. The way we have achieved this is by not only continuing analytically the time coordinate as in OS, but also rotating spinor indices (with the matrix S) . Finally let
P. uan Nieuwenhuizen, A. Waldron / Physics Letters B 389 (19%) 29-36
34
US observe that Zumino’s Euclidean pseudoscalar kinetic action +1/2( 8pB)2 is not damped in the Euclidean path integral (recall that expi Jd4xLM + exp / d4x&). In this light one may prefer to trade the requirement of hermiticity (of the action and supersymmetry transformations) for damping by introducing a field Bf, = iBE.
4. Majorana
and Weyl spinors
Within a path integral approach, Nicolai [ 1 l] obtained an action for Majorana spinors depending only on a single spinor ti (and not $I+) without any reality condition, &@‘faJ)
1 = -&‘E
T
c(Y;+
+ m)tlrE,
(32)
where C is the Minkowski conjugation matrix and his Dn-ac matrices YE = (Y’ , y2, y3, y”) undergo no rotation. In this framework, he is still able to write down a supersymmetric action since, within an N = 1 supersymmetric model, the reality of en is not needed to verify invariance under supersymmetry transformations. Further, the various Fierz identities required are unchanged since the charge conjugation matrix C is still that of the Minkowski theory. In our approach to Majorana spinors we begin with the usual Minkowski action &(Maj)
= -$$TC
+m>$.
= -$@STCSS-1(yP6’P+rn)S$E
= -$tffETcdY
%$+ m)+E
(34)
CE = STCS also satisfies the relations CE = -Cz and CnYgC<’ = -YEPT. Formally this action is the same as that of Nicolai except that Nicolai performs no rotation on spinor indices (and therefore his charge conjugation matrix is the original Minkowski charge conjugation matrix). Neither action is hermitian and in neither case are the classical fields $ and rJ+ related by a reality condition. In an expanded version we shall work through the example of the N = 1 Wess Zumino model. and the matrix
of?, &L.R
‘kL’L,R= +;
1 &y5 X;,R =Xt-’
2’
/&ys 7
= f+L.R
;
= &X+L,R .’
(35) (36)
where @ and x+ are unconstrained Dirac spinors. Transforming (/I ---f S$n and xt + x$ (as usual S = eY4Ys.rr/4)in accordance with our Wick rotation rules, we obtain the transformation rules for Weyl spinors
(33)
Dropping the reality constraint, we can now unambiguously perform our Wick rotation on r/~ as given above. We find &(Maj)
In Minkowski spacetime one may always rewrite a Weyl spinor as a Majorana spinor. However, we found it necessary to drop the reality condition on Majorana spinors when continuing to Euclidean space. The Berezin integration of the path integral is not sensitive to this modification and in this way we were able to reproduce continued Greens functions. We now consider the Wick rotation for Weyl spinors. In Minkowski spacetime$t_=~(1+$)$and@~=$+~(1+y5> is the hermitian conjugate of (GIL.As suggested by the analysis for Majorana spinors, we should first drop this hermiticity requirement and then perform the Wick rotation as for Dirac spinors. Therefore we consider spinors $ and x+, unrelated by hermitian or complex conjugation conjugation but nevertheless eigenstates
where we used s-’
1 iY5 Fs=y
l&Y;
(and similarly for S and S-’ interchanged). Consider now the Minkowski action for a left-handed Weyl spinor (in which we have already relaxed hermiticity by replacing $+ ---f xt) &(Weyl,
I.,) = -x~Y”##L
= -X+Y4# Fe. ’ + y5 (39)
Under Wick rotation we obtain &(Weyl,L)
= X~Y~(“~@s
= -XEt#E#EL. (40)
P. van Nieuwenhuizen, A. Waldron /Physics
where in Euclidean space
As in the the Majorana case this action is not (and cannot be) hermitian. Further notice that the x[ * &, coupling of Minkowski space becomes xi H $r_ in Euclidean space [ lo]. In an expanded version we shall apply this approach to N = 1 super Yang Mills theory. 5. Conclusion
We have defined a continuous Wick rotation on Dirac spinors $ and their hermitian conjugates by means of a Lorentz transformation S(0) in the Euclidean-Minkowski tr plane. The resulting Euclidean Greens functions are not only obtained from their Minkowski counterparts by continuation in the time variable, but also by a rotation of spinor indices. In this respect our work agrees with work of Schwinger, who, however, did not implement this idea on the basic fields. For Majorana and Weyl spinors however, one is forced to drop the reality/hermiticity condition if one wants to perform the Wick rotation and hence our formalism, which still uses the Lorentz rotation S, now resembles the Osterwalder-Schrader approach. Our final rule for the Wick rotation on spinors is $Vr,x)
4 S(@$e(r,x),
(41)
ICl+(tJ) h &rJ)s(e)
(42)
with S(6) = ,Y4YSs/*.Using s(fl)y4 = y4S-’ (d), the Dirac Minkowski action goes over into
j
-‘&4(‘&
3 k + ?@4
+ m)@E
(43)
and since we found that r4 = -$, the action is clearly hermitian and SO(4) invariant. Thus the troublesome 7” is not omitted in Euclidean space as one might expect, but rather reinterpreted as gE in Euclidean space [lo]. With the understanding of the Wick rotation provided by our formalism one can explicitly and continuously follow how vertices and propagators in in-
Letters B 389 (1996) 29-36
35
teracting field theories change as one rotates from the Minkowski to the Euclidean case. This has practical advantages, for example it explains which terms with factors of 9 and axial vector fields in regulators for anomalies acquire factors of i. Another useful application is supersymmetry; our rules automatically generate N = 1 and N = 2 theories in Euclidean space whenever they exist in Minkowski space. Also, another obvious open question is how to generalize our Wick rotation to odd dimensions in which there is no “g”. One possibility is to double the Dirac matrices, which suggests that the cases in which one is able to operate with undoubled spinor degrees of freedom in Euclidean space are the exception rather than the norm [ 181. In this article we have discussed only the path integral aspects of the Wick rotation. The OsterwalderSchrader approach is entirely based on a canonical approach, and our Wick rotation can also be formulated in a canonical way, but this will be published elsewhere [ 181. There we also discuss the OsterwalderSchrader reflection positivity axiom which is the main ingredient for establishing the existence of a positive semi-definite self-adjoint Hamiltonian lo . The last comment we wish to make is perhaps the most interesting but certainly the least understood and speculative. Both in our work and also in that of Osterwalder-Schrader, one keeps noticing five-dimensional aspects tt . For example, in the OS canonical approach, the expansion into creation and annihilation operators has a normalisation factor (k* + m*>-l/* in Euclidean space, as opposed to (k’+m*)-I/* m ’ Minkowski spacetime. The former is clearly the usual normalization of a five-dimensional theory at t = 0, with four space momenta k;. Further Schwinger introduces new fields B(x) for the original fields A(x) which satisfy the commutation loIn more detail, this axiom requites that there exists an antilinear mapping 0 which transforms an arbitrary function F of the fields at positive times into a function OF of fields at negative times such that (OFF) is semi-positive. If such a mapping exists one can define a Hiibert space with positive norm and a positive transfer matrix. The choice of time axis with respect to which time is defined is arbitrary and breaks the four-dimensional symmetry but the results do not depend on the axis chosen [ 191. I1 It may be interesting also to recall the work of [ 201 in which the four-dimensional Wess Zumino model is obtained by dimensional reduction “by Legendre transformation” from the on-shell theory five dimensions
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P. oan Nieuwenhuizen. A. Waldron /Physics
relations [A(x),B(y)] = #(x - y), which again can be viewed as an equal time canonical commutation relation in five dimensions. We found a Wick rotation from Minkowski to Euclidean space which is a five-dimensional Lorentz rotation, but there are many more questions and ideas yet to be developed. Acknowledgements It is a pleasure to thank R. Schrader, P Breitenlohner, D. Maison, E. Seiler, C. Preitschopf and B. Zumino for discussions.
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