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A Wick rotation for spinor fields: the canonical approach
13 August 1998
Physics Letters B 433 Ž1998. 369–376
A Wick rotation for spinor fields: the canonical approach Andrew Waldron
1
Institute for Theoretical Physics, State UniÕersity of New York at Stony Brook, Stony Brook, NY 11794-3840, USA Received 27 February 1997; revised 2 April 1998 Editor: M. Dine
Abstract Recently we proposed a new Wick rotation for Dirac spinors which resulted in a hermitean action in Euclidean space. Our work was in a path integral context, however, in this note, we provide the canonical formulation of the new Wick rotation along the lines of the proposal of Osterwalder and Schrader. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction and review Field theories Wick rotated to Euclidean space are the subject of an enormous body of research. In particular, modern non-perturbative studies of supersymmetric theories Žfor example instantons and the study of Donaldson invariants for compact manifolds. depend on the introduction of Euclidean field theories. Clearly then, it is of crucial importance to understand how one performs a Wick rotation for spinors. In a previous publication in this journal w1x we observed that there existed two apparently distinct approaches to Dirac spinors in Euclidean space. Namely the approach of Osterwalder and Schrader w3x ŽOS. in which the fields c and its Dirac conjugate c are taken to be independent and hermiticity is forsaken and the approach of Schwinger w4x and later Zumino w5x in which spinor degrees of freedom are
1
E-mail: [email protected]. Address as of 9r1r1996: NIKHEF, Postbus 41882, 1009 DB Amsterdam, The Netherlands. E-mail: [email protected].
undoubled and the action in Euclidean space is hermitean. Within a path integral context, the distinction between integrating over fields c and c † versus independent fields c and x † Ž c / x . is only semantic due to the algebraic nature of Grassmann integration, so the real puzzle was therefore to understand how Schwinger was able to maintain hermiticity whereas OS did not. ŽWe stress, however, that ultimately the requirement of hermiticity is secondary to the requirement that Euclidean Greens functions reproduce analytically continued Greens functions of the Minkowski theory. Nevertheless, we wish to elucidate the relation between these alternative formulations of Euclidean Quantum Field Theory 2 .. This problem was solved by introducing a new Wick rotation for Dirac spinors which acted only on the fundamental fields and coordinates. For vectors AmŽ t, x ., the Wick rotation to Euclidean space is performed by transforming both the
2
A geometric understanding of our new Wick rotation has been provided in terms of a dimensional reduction along the time direction w6x.
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 5 9 6 - 6
A. Waldronr Physics Letters B 433 (1998) 369–376
370
time coordinate t ™ it and vector indices by a matrix Vmn s diagŽ i,1,1,1., i.e., AmŽ t, x . ™ Vmn AmE Žt , x .. However for complex vectors the complex conjugate A†m transforms under the same matrix Vmn rather than Vmn †. This observation led us to introduce the following Wick rotation for Dirac spinors w1x:
c Ž t , x . ™ S Ž u . cu Ž tu , x .
Ž 1.
c † Ž t , x . ™ cu† Ž tu , x . S Ž u .
Ž 2.
yi u
t™e
where
tu ,
Ž 3. ,
u g w 0,pr2 x .
so that our Wick rotation induces a similarity transformation upon the Dirac matrices in Dirac bilinears cGA c for some combination of Dirac matrices GA ,
g Ž u . ' Sy1 Ž u . g S Ž u . s g ' g E
Ž 6.
g 4 Ž u . ' Sy1 Ž u . g 4 S Ž u . s g 4 cos u q g 5 sin u g Žu . 'S
y1
5
4
Ž 7.
5
Ž u . g S Ž u . s yg sin u q g cos u . Ž 8.
Note then that gu4s p r2 ' g E4 s g 5, gu5s p r2 ' g E5 s
Our conventions are as follows, the Minkowski Dirac matrices g m s Žg 0 'y ig 4 ,g . m, where g 4 and g are hermitean and g m ,g n 4 s 2 h mn s 2diagŽy1,1,1,1. mn . The matrix g 5 ' g 1g 2g 3g 4 is hermitean. 4 Note that Lorentz invariance of the intermediate interpolating theories is obtained only if one complexifies so that cu and cu† are independent spinors and hermiticity of the interpolating Dirac action is lost. Our claim is not that c and c † remain dependent under Wick rotation, but rather, that for Dirac spinors, hermiticity at the endpoint Žthe Euclidean theory. may be regained. In this sense one may think of hermiticity as a symmetry property of the Euclidean theory.
g 5 u r4
and the troublesome g 4 in c has been reinterpreted as yg E5 in the Euclidean theory. Applying this Wick rotation to the action for a free massive Dirac spinor i "
1
SM ' y
"
Hdtdx c
= g0
ž
Ž 4.
c Ž t , x . ' c † Ž t , x . g 4 ™ cu† Ž tu , x . g 4 Sy1 Ž u . , Ž 5 .
4
Ž 9.
E g 4 g 5 u r2
The matrix SŽ u . is unitary and the parameter u is introduced to provide a continuous 4 interpolation between the Minkowski and Euclidean theories. At the initial value u s 0, SŽ u s 0. s I and cus0 ' c , cu†s0 ' c † and tus0 ' t ' x 0 ' yx 0 take their usual Minkowski values, whereas at the endpoint u s pr2, 4 5 S Ž u s pr2. s e g g p r4 ' S and cus p r2 ' c E , † † cus p r2 ' c E and tus p r2 ' t ' x 4 ' x 4 are their Euclidean counterparts. Observe that S Ž u .g 4 s g 4 Sy1 Ž u . whereby
3
c Ž t , x . s c † Ž t , x . g 4 ™ yc E† Ž t , x . g E5 ey g
3
SŽ u . se
5
yg 4 whereby the Wick rotation of the Dirac conjugate spinor c yields
Et
†
Ž t, x.g 4
qgP≥qm c Ž t, x. ,
/
Ž 10 .
we obtain 1 y "
SE '
1 "
Hdt dx c E
= g4
ž
Et
† E
Ž t , x . g E5
qgP≥qm c Žt , x . .
/
Ž 11 .
This action is hermitean, SO Ž4. invariant and is the result of a Wick rotation acting as an analytic continuation t ™ yit and a simultaneous rotation on spinor indices. The above Wick rotation produces a Euclidean theory for spinor fields whose action, in the exponent of a Euclidean path integral, yields Euclidean Greens functions which are related to the usual Minkowski Greens functions by analytic continuation and a rotation on spinor indices by the matrix S introduced above. However, along with necessary and sufficient conditions for a Euclidean theory to produce the analytically continued counterparts of the Greens function of a given Minkowski theory w2x, Osterwalder and Schrader w3x have explicitly constructed a canonical Euclidean theory in terms of Euclidean Dirac spinor fields acting in a Euclidean Fock space whose Greens functions are the analytic continuations of the corresponding Minkowski Greens functions. In the remainder of this note, we shall reformulate their work in the context of our Wick rotation for Dirac spinors.
2. The main ingredients of the OS construction We begin by briefly sketching the main ingredients of the OS construction. Firstly we analytically
A. Waldronr Physics Letters B 433 (1998) 369–376
continue canonical Minkowski fields to imaginary times t ™ yit . The canonical ŽHeisenberg. Minkowski field satisfying the free Dirac equation Ž Euq m. c s 0 is given by
c Ž t , x . s e i H 0 tc Ž 0, x . eyi H 0 t ,
Ž 12 .
The continuation of the Dirac conjugate field c is defined in the same way
c Ž yit , x . s e H 0tc † Ž 0, x . ig 0 eyH 0t d3k
with the usual mode expansion
s
dk
H Ž 2p . Ž b P u k
3r2
ke
i kP x
and normal ordered free Hamiltonian H 0 s Hdk v k Ž b†k P b k q d k† P d k . where v k s Ž k 2 q m2 .1r2 . We denote the sum over spin polarizations by a dot, i.e., b k P u k s Ý rs1,2 b kr P u kr . The orthonormal spinor wave functions satisfy Ž iku q m. u k s 0 s Žyiku q m. Õyk and spin polarization sums yiku q m
uk P uk s
2 vk
;Õyk P Õyk s
yiku y m 2 vk
.
Ž 14 .
Defining the usual Minkowski vacuum <0: via d k <0: s 0 s b k <0: and imposing commutation relations for the modes
b k ,b†k 4 s d 3 Ž k y k . 1 s d k ,d k† 4 X
X
X
H Ž 2p .
s yi
4
yiku q m k 2 q m2 y i e
e
i kmŽ xyy. m
'DŽ xyy. .
Ž 16 .
The analytically continued Minkowski fields are constructed by allowing the Minkowski field Ž12. to undergo imaginary time evolution with t s yit ,
c Ž yit , x . s e H 0tc Ž 0, x . eyH 0t d3k
H(
Ž 2p .
†
3
= b k P u k eyv ktqi kP x q d k† P Õyk e v ktyi kP x 4
Ž 17 .
ig 0 .
Ž 18 .
It is important to realize from the last equality in Ž18. that hermitean conjugation and analytic continuation do not commute, but rather an additional ‘‘Euclidean time reversal’’ t ™ yt is required. The concept of reflection positivity follows from this remark. The continued free two-point function, or propagator, can now straightforwardly be constructed from the continued Minkowski fields, D Ž yit q i s , x y y . ' ²0 < T˜c Ž yit , x . c Ž yi s , y . <0: s u Ž t y s . ²0 < c Ž yit , x . c Ž yi s , y . <0: y u Ž s y t . ²0 < c Ž yi s , y . c Ž yit , x . <0: s
dk 4 d 3 k yiku OS q m
H Ž 2p .
4
d E2 q m2
' DE Ž x E y y E . .
²0 < Tc Ž x . c Ž y . <0: dk 0 d 3 k
s c Ž x ,q it .
Ž 15 .
Žwe have suppressed the polarization indices r, s s " so 1 denotes d r s . one obtains the two-point function Žpropagator.
3
= b†k P u k eq v ktyi kP x q d k P Õyk ey v ktqi kP x 4
q d k† P Õyk eyi kP x .
Ž 13 .
s
H(
Ž 2p .
c Ž 0, x . s
371
E
E
e i km Ž xyy.m
Ž 19 .
Let us make a few comments. The symbol T˜ denotes time ordering, but now with respect to t , the continued time. Further, ku OS ' g 4 k 4 q g P k so that spinor indices undergo no rotation in the OS approach, rather an i is simply ‘‘borrowed’’ from the relation g 0 s yig 4 . Also kmE xmE ' k 4t q k P x is the Euclidean inner product. Furthermore, it is easy to show that this result for D Žyit , x . is exactly the same as that obtained from the function D Ž t, x . in Ž16. by a direct Žunique. analytic continuation in the time variable. So far we have done nothing except consider the usual Minkowski fields acting in the Minkowski Fock space but at imaginary values of the time coordinate. However in the last line of Ž19. we denoted D Žyit , x . s DE Ž x E . because the next step is to construct Euclidean fields acting in a Euclidean
A. Waldronr Physics Letters B 433 (1998) 369–376
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Fock space whose two-point function is DE Ž x E .. General Greens functions can be reconstructed via use of the Wick theorem. The time coordinate should play no preferred role ˆ in Euclidean space so the OS fields anticommute at all points x E and y E Ž x E and y E are now, of course, four dimensional.. Hence there is no time ordering in their Euclidean formulation and the Euclidean fields are expanded in terms of modes depending on Euclidean four-momenta. In any case there are no plane wave solutions to ŽyI E q m2 . c E s 0 in flat Euclidean space, so that, in the OS construction, Euclidean fields are off-shell. Hence their Euclidean fields have a form which is rather close to that of a five-dimensional Minkowski field at zero time. Explicitly the Euclidean canonical Dirac spinor of OS is given by
tion of hermitean conjugation is replaced by the composition of hermitean conjugation and a unitary involution Q defined as follows. For Bose fields f E Žt , x . Žwhich may be easily be treated in the OS approach without doubling, see w3x for details. Q acts simply as Euclidean time reversal,
cEŽ x.
x E† Ž x . d4 k
s
H
(Ž 2p . V
Bk ( Uk e i k x q D†k ( Vyk eyi k x 4 ,
4
Qf E Ž t , x . Qy1 s f E Ž yt , x . s f E Ž Q x . ,
Ž 22 .
where Q Žt , x . s Žyt , x .. This definition is motivated by the remark above that for continued Minkowski fields, one needed an additional Euclidean time reversal when comparing imaginary time evolution of the hermitean conjugated field with the hermitean conjugate of the imaginary time evolved field Žsee Ž18... For spinors however, the action of Q is more subtle. Define a field x E† Ž x ., the Euclidean analogue of the Minkowski field c Ž x,t ., by d4 k s
k
H
(Ž 2p . V
† eik x 4 . Bk† ( Wk†eyi k x q Dk ( Xyk
4
k
Ž 20 .
Ž 23 .
where V k ' Ž k 2 q m 2 .1r2 ,e i k x s e iŽ k 4tqkP x . and from now one we drop the subscript E on the Euclidean four-vectors k and x. The mode operators Bk and D†k act in a Euclidean Fock space with vacuum <0: E defined such that Bk <0: E s 0 s D k <0: E . The only non-vanishing anticommutation relations of the modes now give a four dimensional delta function,
Since we already doubled the number of spin polarizations Ž R,S s 1, . . . ,4., it would be a redoubling if the field x E† Ž x . were independent 5 of c E Ž x .. Rather, x E† is related to c E by both hermitean conjugation and the action of Q ,
Bk , Bk† 4 s d 4 Ž k y k . 1 s Dk , D†k 4 , X
x E† Ž t , x . ' Qy1c E† Ž yt , x . ig 0Q s Qy1c E† Ž Q x . ig 0Q .
Ž 21 .
Ž 24 .
where we again suppress polarization indices so that 1 denotes d R S. However, let us stress that the indices R and S no longer run over values 1,2, rather it is necessary to double the spin polarization degrees of freedom whereby R,S s 1, . . . ,4, and we denote Bk ( Uk ' Ý4Rs1 BkR UkR. The spinor wave functions Uk and Vyk do not satisfy any equations of motion. At this point the fields c E Ž x . in Ž20. and c Žyt , x . in Ž17. are totally unrelated, they act in different Fock spaces. The next task is to construct a conjugate momentum field to c E Ž x . Ži.e., the analogue of c † in the Minkowski case.. In the OS proposal, the answer is no longer w c E Ž x .x† Žalthough a key feature of the canonical formulation of our new Wick rotation is that this property will be retained., instead the opera-
The relation Ž24. together with the expansions Ž23. and Ž20. imply that Q has a more complicated action on the modes Žand in turn states in the Euclidean Fock space. such that
X
X
Qy1 BQ† k ( UQ†k ig 0Q s Bk† ( Wk† ; † Qy1 DQ k ( Vy† Q k ig 0Q s D k ( Xyk ,
Ž 25 .
where Q k s Žyk 4 , k .. Given the explicit forms Žsee w3x. of the spinor wave-functions Uk ,Vyk ,Wk and 5
Contrast this to an OS path integral approach for Dirac spinors in which the ‘‘doubling’’ of spinor degrees of freedom in Euclidean space is introduced by taking the field x E† to be independent of c E .
A. Waldronr Physics Letters B 433 (1998) 369–376
Xyk one can write down the action of Q on the mode operators Bk and Dk . For our purposes it is enough to note that the spinor wavefunctions satisfy spin polarization sums constructed such that one obtains the desired two-point function in Ž19., yiku Uk ( Wk† s
OS
qm
Vk
yiku † ;Vyk ( Xyk s
OS
ym
Vk
.
Ž 26 . Defining Q <0: E s <0: E one can then also calculate the action of Q on states in the Euclidean Fock space. The mode expansions Ž20. and Ž23. along with the spin polarization sums in Ž26. yield
c E Ž x . , x E† Ž y . 4 s 0 .
Ž 27 .
Furthermore, using Ž20., Ž21., Ž23. and Ž26. it is easy to verify the following equalities: DE Ž x y y . 'E ²0 < c E Ž x . x E† Ž y . <0: E s yE ²0 < x E† Ž y . c E Ž x . <0: E s D Ž yit q i s , x y y . , Ž 28 . Ž . Ž . Ž where De x y y in 28 agrees with D E x y y . in Ž19. so that the Euclidean two-point function without time ordering reproduces the continued, time ordered Minkowski two-point function. The final ingredient is the relation between states in the Euclidean Fock space, and those in the physical Minkowski Hilbert space. This is provided by the following mapping W : < X : E ™ < WX : M , Ž 29 . from an arbitrary state < X : E in the Euclidean Fock space to some state < WM : M in the Minkowski Hilbert space. We shall call this mapping the ‘‘OS-Wick map’’ and it is defined as follows. A general state in the Euclidean Fock space w3x may be represented as
the right. The OS-Wick map may now be defined by its action on the state < X : E in Ž30., W < X : E s < WX : M ' : c Ž yit 1 , x 1 . . . . c Ž yitmqn , x mqn . : <0: . Ž 31 . where, for brevity, we have suppressed the smearing by test functions f i Ž x i .. The fields c Žyit 1 , x 1 . and c Žyitmq n , x mq n . are precisely the continued Minkowski fields defined in Ž17. and Ž18., respectively, above. In w3x, the following central theorem is proven E
²Q X < Y : E sM ²WX < WY : M .
E
²Q X < X : E G 0 .
Ž 33 .
As yet we have made no mention of how dynamics are included in this proposal, but at this point we refer the reader to the original work of Osterwalder and Schrader w3x. Let us now give the generalization of the above construction to include our new Wick rotation.
3. The canonical formulation of the new Wick rotation In w3x, it is argued that the field x E† cannot be replaced by c E† since the two-point function, ²0 < c E Ž x . x E† Ž y . <0: E d4 k
The functions f i Ž x i . are some choice of test functions and it is convenient to normal order this expression as denoted by :: by which we mean all annihilation operators Bk and D k are to be pulled to
Ž 32 .
which states that inner products of states in the Euclidean Fock space are related to those in the Minkowski Hilbert space by the OS-Wick map and the unitary involution Q . The inner product in the Minkowski Hilbert space should be positive definite, whereby we immediately obtain the OS reflection positivity condition
E
Ž30.
373
s
H Ž 2p .
yiku OS q m 4
k E2 q m2
e i k E Ž xyy. E ,
Ž 34 .
is then inconsistent because only the left hand side is invariant under hermitean conjugation and the interchange of x and y. In light of our new Wick rotation, the remedy is obvious. One should replace ku OS by ku E s k Emg Em where the matrices g Em are defined in the introduction Žand were obtained via a
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similarity transformation induced by the rotation of spinor indices. and put
x E† s yc E† g E5 .
Ž 35 .
In order to incorporate our rotation of spinor indices in the OS proposal, we replace the OS-Wick map W ˜ defined as follows: by a new OS-Wick map W,
˜ :M , W˜ : < X : ™ < WX
cEŽ x. s
Ž 36 .
d4 k
H(
Ž 2p .
4
Bk e i k x q Dk) eyi k x 4 ,
Ž 41 .
from which it follows that
where, in general, < X : E s :c E Ž x1 . . . .
four values. The necessity to reproduce the continued 3 q 1 dimensional Minkowski propagator, however, can be used to rule out directly replacing c E and x E† by their zero time, five dimensional counterparts. In this light, we consider the more general ansatz
c E†
Ž x nqm . Ž
yg E5
. : <0: E , Ž 37.
yc E† Ž x . g E5 d4 k
and
H(
Bk†g E5 eyi k x q Dki g E5 e i k x 4 ,
sy
˜ : M s :Sc Ž yit 1 , x 1 . . . . < WX
Ž 2p .
y1
: <0: M ,
4
Ž 38 .
Ž 42 .
with S s e . Hence, Euclidean Greens functions are related to their continued Minkowski Greens functions by an additional rotation of spinor indices. For example, for the free two-point function, we find
where we have replaced the combinations Uk ( Bk and D†k ( Vyk by the operator-valued four-spinors Bk and Dk) , respectively. We still require that Bk <0: E s 0 s Dk <0: E . By virtue of the rotation on spinor indices, the unitary involution Q is now defined on spinors in the same way as for bosons
=c Ž yitnqm , x nqm . S g 4 g 5p r4
E
²0 < c E Ž x . c E† Ž y . Ž yg E5 . <0: E d4 k s
H Ž 2p .
yiku E q m 4
k 2 q m2
e i kŽ xyy.
s Sy1 D Ž yit q i s , x y y . S .
Ž 39 .
Qc E Ž x . Qy1 s c E Ž Q x . ;Qc E† Ž x . Qy1 s c E† Ž Q x . , Ž 43 .
We must now construct the field c E , the Euclidean Fock space and the unitary involution Q . Let us briefly mention two unfruitful avenues before we give our solution. The first would be to require Wk† in Ž23. and Uk in Ž20. to satisfy
which in turn defines the action of Q on the spinor modes
Wk† s yUk†g E5
Clearly this action is involutive. Euclidean ultralocality,
Ž 40 .
Žalong with an analogous condition on Vyk and Xyk .. However, we must still reproduce the correct continued Minkowski two-point function which requires Uk ( Wk† s yiku E q m for which there is no solution when Ž40. holds 6 . A second, bolder proposal would be to notice that the expressions Ž20. and Ž23. are reminiscent of those of a five Ž5 s 1 q 4. dimensional Dirac spinor at time t 5 s 0, except that the polarization sums denoted by ‘‘(’’, should run over two, instead of
Q Bk†Qy1 s BQ† k , Q
Dki Qy1 s DQik
.
Ž 45 .
†
0 s c E Ž x . ,y c E Ž y . g E5 4 ,
Ž 46 .
is satisfied by requiring ) i , Dyk Bk , Bk† 4 s y Dyk 4. X
X
Ž 47 .
The correct two-point function as in Ž39. is ensured by the following anticommutation relations for the ‘‘spinor modes’’ y Bk , Bk†Xg E5 4 s
6 To see this, take a basis in which g E5 sdiagŽ1,1,y1,y1. and the other Dirac matrices are off-diagonal. Multiplying by yg E5 and tracing yields UkR ' 0Ž Rs1, . . . ,4..
Ž 44 .
E
yiku E q m
V k2
d 4 Ž k y kX . .
Ž 48 .
m .Žyg E5 . is hermitean with eigenThe matrix Ž yikuV kq 2 values "V k and may be diagonalized Žin a basis for
A. Waldronr Physics Letters B 433 (1998) 369–376
the Dirac matrices in which g E5 is diagonal. by defining 1 Bk s
(V
Uk B˜k , k
(2 Ž V y m . V k
The combination
k
†
s Ž Uy1 k . s Ž Uyk .
y1
½ B˜ , B˜ 5 s yg †X k
1 'V Uk is the analogue of the OS k
5 4 Ed
Ž k y kX . .
cEŽ x.
˜ (Ž 2p . V ½U B e k
4
k
ik x
q Uyk D˜ k) eyi k x ,
5
k
Ž 52 .
) i X , D˜ yk Ž k y k X . s y ½ D˜ yk 5, Ž 54 .
c E Ž x . , x E† Ž y . 4 ' y c E Ž x . , c E† Ž y . g E5 4 s 0 , Ž 55 . and possess the desired two-point function,
Ž 51 .
That the Euclidean Fock space now contains negative norm states causes no problems since we only require positiÕe norm in the Minkowski Hilbert space, which is assured since, by construction, our fields satisfy the same axioms as those of the OS proposal. Let us elaborate on this point since it constitutes one of the main observations of this note. The OS reflection positivity condition Ž33. is a statement about certain inner products in the Euclidean Fock space deriÕed from the positivity of inner products in the Minkowski Hilbert space. In the OS proposal, a Euclidean Fock space is employed in which the mode operators are doubled and all norms are positive but hermiticity, Ži.e., the relation between a spinor and its Dirac conjugate. are forsaken. Our observation is that one may trade positivity of norms in the Euclidean Fock space in lieu of hermiticity. This is possible since the Euclidean Fock space is doubled and indeed we find that exactly half of the Euclidean mode operators yield states having negative norm. To continue, we may ‘‘diagonalize’’ the Dk modes in a similar fashion so that the final result for our Euclidean fields reads
5 4 Ed
By construction these fields anticommute at all points in Euclidean space
E
d4 k
†X k
.
k
spinor wave function Uk . In this basis the mode relations now read
H
Ž 53 .
½ B˜ , B˜ 5 s yg Ž 50 .
s
x E† Ž x . s yc E† Ž x . g E5 s yQy1c E† Ž Q x . g E5 Q . The modes satisfy commutation relations
V k y m q iku E
k
and
Ž 49 .
where Uk s
375
²0 < c E Ž x . c E† Ž y . Ž yg E5 . <0: E d4 k s
H Ž 2p .
yiku E q m 4
k E2 q m 2
e i k E Ž xyy. E .
Ž 56 .
which is consistent with hermiticity. We have now reproduced the building blocks ŽŽ56., Ž54., Ž43. and Ž38.. of the OS construction and the rest of their proposal may now be inherited unaltered except for the replacement everywhere of the field x E† by yc E† g E5 and the extra rotation of spinor indices performed by the new OS-Wick map.
4. Conclusion In this note we have presented a generalization of the canonical work of Osterwalder and Schrader for Dirac spinors which fuses their approach with the Schwinger–Zumino w4,5x approach in which hermiticity is maintained although we still found it necessary to double the set of Euclidean fermionic mode operators, half of which now yield states having negative norm. Nonetheless, OS positivity is maintained, since, necessarily, Minkowski states have positive norm and the new proposal reproduces the same fundamental axioms as that of the original OS proposal. The essential new ingredient is our new Wick rotation for fermions under which spinor indices also rotate w1x. We would also suggest that our generalization adds to the formal simplicity of the OS construction.
376
A. Waldronr Physics Letters B 433 (1998) 369–376
Finally, one may wonder what happens in the case of Majorana or Weyl spinors. In w1x we found that for Majorana and Weyl spinors in four dimensions that the requirement of hermiticity must be dropped. The extension of the work of OS to Majorana spinors was given by Nicolai w7x who noted that although there existed no consistent reality condition for Majorana spinors in four dimensional Euclidean space, one could nonetheless define a symplectic reality condition on the mode operators Bk and Dk following which the OS proposal may also be simply inherited. It is not a difficult matter to apply the generalization we have given above also to Nicolai’s work, and although one cannot regain hermiticity, the same formal algebraic simplifications as above occur. One may then even study N s 1 supersymmetric systems. As usual real bose fields undergo complex supersymmetry transformations in Euclidean space since the canonical supersymmetry charge Q no longer satisfies any reality condition. Such considerations may also be formulated in superspace Žsee also w8x..
Acknowledgements I am deeply indebted to Peter van Nieuwenhuizen for numerous detailed discussions and suggestions.
References w1x P. van Nieuwenhuizen, A. Waldron, Phys. Lett. B 389 Ž1996. 29; A Continuous Wick Rotation for Spinor Fields and Supersymmetry in Euclidean Space, in: Proceedings of Gauge Theories, Applied Supersymmetry and Quantum Gravity, London, World Scientific, 1996. w2x K. Osterwalder, R. Schrader, CMP 31 Ž1973. 83; CMP 42 Ž1975. 281. w3x K. Osterwalder, R. Schrader, Phys. Rev. Lett. 29 Ž1972. 1423; Helv. Phys. Acta 46 Ž1973. 277. w4x J. Schwinger, Phys. Rev. 115 Ž1959. 721. w5x B. Zumino, Phys. Lett. B 69 Ž1977. 369. w6x M. Blau, G. Thompson, Phys. Lett. B 415 Ž1997. 242. w7x H. Nicolai, Nucl. Phys. B 140 Ž1978. 284. w8x K. Schrader, in: Proceedings of the Conference on Advances in Dynamical Systems and Quantum Physics, Capri, World Scientific, 1993.
Physics Letters B 433 Ž1998. 369–376
A Wick rotation for spinor fields: the canonical approach Andrew Waldron
1
Institute for Theoretical Physics, State UniÕersity of New York at Stony Brook, Stony Brook, NY 11794-3840, USA Received 27 February 1997; revised 2 April 1998 Editor: M. Dine
Abstract Recently we proposed a new Wick rotation for Dirac spinors which resulted in a hermitean action in Euclidean space. Our work was in a path integral context, however, in this note, we provide the canonical formulation of the new Wick rotation along the lines of the proposal of Osterwalder and Schrader. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction and review Field theories Wick rotated to Euclidean space are the subject of an enormous body of research. In particular, modern non-perturbative studies of supersymmetric theories Žfor example instantons and the study of Donaldson invariants for compact manifolds. depend on the introduction of Euclidean field theories. Clearly then, it is of crucial importance to understand how one performs a Wick rotation for spinors. In a previous publication in this journal w1x we observed that there existed two apparently distinct approaches to Dirac spinors in Euclidean space. Namely the approach of Osterwalder and Schrader w3x ŽOS. in which the fields c and its Dirac conjugate c are taken to be independent and hermiticity is forsaken and the approach of Schwinger w4x and later Zumino w5x in which spinor degrees of freedom are
1
E-mail: [email protected]. Address as of 9r1r1996: NIKHEF, Postbus 41882, 1009 DB Amsterdam, The Netherlands. E-mail: [email protected].
undoubled and the action in Euclidean space is hermitean. Within a path integral context, the distinction between integrating over fields c and c † versus independent fields c and x † Ž c / x . is only semantic due to the algebraic nature of Grassmann integration, so the real puzzle was therefore to understand how Schwinger was able to maintain hermiticity whereas OS did not. ŽWe stress, however, that ultimately the requirement of hermiticity is secondary to the requirement that Euclidean Greens functions reproduce analytically continued Greens functions of the Minkowski theory. Nevertheless, we wish to elucidate the relation between these alternative formulations of Euclidean Quantum Field Theory 2 .. This problem was solved by introducing a new Wick rotation for Dirac spinors which acted only on the fundamental fields and coordinates. For vectors AmŽ t, x ., the Wick rotation to Euclidean space is performed by transforming both the
2
A geometric understanding of our new Wick rotation has been provided in terms of a dimensional reduction along the time direction w6x.
0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 5 9 6 - 6
A. Waldronr Physics Letters B 433 (1998) 369–376
370
time coordinate t ™ it and vector indices by a matrix Vmn s diagŽ i,1,1,1., i.e., AmŽ t, x . ™ Vmn AmE Žt , x .. However for complex vectors the complex conjugate A†m transforms under the same matrix Vmn rather than Vmn †. This observation led us to introduce the following Wick rotation for Dirac spinors w1x:
c Ž t , x . ™ S Ž u . cu Ž tu , x .
Ž 1.
c † Ž t , x . ™ cu† Ž tu , x . S Ž u .
Ž 2.
yi u
t™e
where
tu ,
Ž 3. ,
u g w 0,pr2 x .
so that our Wick rotation induces a similarity transformation upon the Dirac matrices in Dirac bilinears cGA c for some combination of Dirac matrices GA ,
g Ž u . ' Sy1 Ž u . g S Ž u . s g ' g E
Ž 6.
g 4 Ž u . ' Sy1 Ž u . g 4 S Ž u . s g 4 cos u q g 5 sin u g Žu . 'S
y1
5
4
Ž 7.
5
Ž u . g S Ž u . s yg sin u q g cos u . Ž 8.
Note then that gu4s p r2 ' g E4 s g 5, gu5s p r2 ' g E5 s
Our conventions are as follows, the Minkowski Dirac matrices g m s Žg 0 'y ig 4 ,g . m, where g 4 and g are hermitean and g m ,g n 4 s 2 h mn s 2diagŽy1,1,1,1. mn . The matrix g 5 ' g 1g 2g 3g 4 is hermitean. 4 Note that Lorentz invariance of the intermediate interpolating theories is obtained only if one complexifies so that cu and cu† are independent spinors and hermiticity of the interpolating Dirac action is lost. Our claim is not that c and c † remain dependent under Wick rotation, but rather, that for Dirac spinors, hermiticity at the endpoint Žthe Euclidean theory. may be regained. In this sense one may think of hermiticity as a symmetry property of the Euclidean theory.
g 5 u r4
and the troublesome g 4 in c has been reinterpreted as yg E5 in the Euclidean theory. Applying this Wick rotation to the action for a free massive Dirac spinor i "
1
SM ' y
"
Hdtdx c
= g0
ž
Ž 4.
c Ž t , x . ' c † Ž t , x . g 4 ™ cu† Ž tu , x . g 4 Sy1 Ž u . , Ž 5 .
4
Ž 9.
E g 4 g 5 u r2
The matrix SŽ u . is unitary and the parameter u is introduced to provide a continuous 4 interpolation between the Minkowski and Euclidean theories. At the initial value u s 0, SŽ u s 0. s I and cus0 ' c , cu†s0 ' c † and tus0 ' t ' x 0 ' yx 0 take their usual Minkowski values, whereas at the endpoint u s pr2, 4 5 S Ž u s pr2. s e g g p r4 ' S and cus p r2 ' c E , † † cus p r2 ' c E and tus p r2 ' t ' x 4 ' x 4 are their Euclidean counterparts. Observe that S Ž u .g 4 s g 4 Sy1 Ž u . whereby
3
c Ž t , x . s c † Ž t , x . g 4 ™ yc E† Ž t , x . g E5 ey g
3
SŽ u . se
5
yg 4 whereby the Wick rotation of the Dirac conjugate spinor c yields
Et
†
Ž t, x.g 4
qgP≥qm c Ž t, x. ,
/
Ž 10 .
we obtain 1 y "
SE '
1 "
Hdt dx c E
= g4
ž
Et
† E
Ž t , x . g E5
qgP≥qm c Žt , x . .
/
Ž 11 .
This action is hermitean, SO Ž4. invariant and is the result of a Wick rotation acting as an analytic continuation t ™ yit and a simultaneous rotation on spinor indices. The above Wick rotation produces a Euclidean theory for spinor fields whose action, in the exponent of a Euclidean path integral, yields Euclidean Greens functions which are related to the usual Minkowski Greens functions by analytic continuation and a rotation on spinor indices by the matrix S introduced above. However, along with necessary and sufficient conditions for a Euclidean theory to produce the analytically continued counterparts of the Greens function of a given Minkowski theory w2x, Osterwalder and Schrader w3x have explicitly constructed a canonical Euclidean theory in terms of Euclidean Dirac spinor fields acting in a Euclidean Fock space whose Greens functions are the analytic continuations of the corresponding Minkowski Greens functions. In the remainder of this note, we shall reformulate their work in the context of our Wick rotation for Dirac spinors.
2. The main ingredients of the OS construction We begin by briefly sketching the main ingredients of the OS construction. Firstly we analytically
A. Waldronr Physics Letters B 433 (1998) 369–376
continue canonical Minkowski fields to imaginary times t ™ yit . The canonical ŽHeisenberg. Minkowski field satisfying the free Dirac equation Ž Euq m. c s 0 is given by
c Ž t , x . s e i H 0 tc Ž 0, x . eyi H 0 t ,
Ž 12 .
The continuation of the Dirac conjugate field c is defined in the same way
c Ž yit , x . s e H 0tc † Ž 0, x . ig 0 eyH 0t d3k
with the usual mode expansion
s
dk
H Ž 2p . Ž b P u k
3r2
ke
i kP x
and normal ordered free Hamiltonian H 0 s Hdk v k Ž b†k P b k q d k† P d k . where v k s Ž k 2 q m2 .1r2 . We denote the sum over spin polarizations by a dot, i.e., b k P u k s Ý rs1,2 b kr P u kr . The orthonormal spinor wave functions satisfy Ž iku q m. u k s 0 s Žyiku q m. Õyk and spin polarization sums yiku q m
uk P uk s
2 vk
;Õyk P Õyk s
yiku y m 2 vk
.
Ž 14 .
Defining the usual Minkowski vacuum <0: via d k <0: s 0 s b k <0: and imposing commutation relations for the modes
b k ,b†k 4 s d 3 Ž k y k . 1 s d k ,d k† 4 X
X
X
H Ž 2p .
s yi
4
yiku q m k 2 q m2 y i e
e
i kmŽ xyy. m
'DŽ xyy. .
Ž 16 .
The analytically continued Minkowski fields are constructed by allowing the Minkowski field Ž12. to undergo imaginary time evolution with t s yit ,
c Ž yit , x . s e H 0tc Ž 0, x . eyH 0t d3k
H(
Ž 2p .
†
3
= b k P u k eyv ktqi kP x q d k† P Õyk e v ktyi kP x 4
Ž 17 .
ig 0 .
Ž 18 .
It is important to realize from the last equality in Ž18. that hermitean conjugation and analytic continuation do not commute, but rather an additional ‘‘Euclidean time reversal’’ t ™ yt is required. The concept of reflection positivity follows from this remark. The continued free two-point function, or propagator, can now straightforwardly be constructed from the continued Minkowski fields, D Ž yit q i s , x y y . ' ²0 < T˜c Ž yit , x . c Ž yi s , y . <0: s u Ž t y s . ²0 < c Ž yit , x . c Ž yi s , y . <0: y u Ž s y t . ²0 < c Ž yi s , y . c Ž yit , x . <0: s
dk 4 d 3 k yiku OS q m
H Ž 2p .
4
d E2 q m2
' DE Ž x E y y E . .
²0 < Tc Ž x . c Ž y . <0: dk 0 d 3 k
s c Ž x ,q it .
Ž 15 .
Žwe have suppressed the polarization indices r, s s " so 1 denotes d r s . one obtains the two-point function Žpropagator.
3
= b†k P u k eq v ktyi kP x q d k P Õyk ey v ktqi kP x 4
q d k† P Õyk eyi kP x .
Ž 13 .
s
H(
Ž 2p .
c Ž 0, x . s
371
E
E
e i km Ž xyy.m
Ž 19 .
Let us make a few comments. The symbol T˜ denotes time ordering, but now with respect to t , the continued time. Further, ku OS ' g 4 k 4 q g P k so that spinor indices undergo no rotation in the OS approach, rather an i is simply ‘‘borrowed’’ from the relation g 0 s yig 4 . Also kmE xmE ' k 4t q k P x is the Euclidean inner product. Furthermore, it is easy to show that this result for D Žyit , x . is exactly the same as that obtained from the function D Ž t, x . in Ž16. by a direct Žunique. analytic continuation in the time variable. So far we have done nothing except consider the usual Minkowski fields acting in the Minkowski Fock space but at imaginary values of the time coordinate. However in the last line of Ž19. we denoted D Žyit , x . s DE Ž x E . because the next step is to construct Euclidean fields acting in a Euclidean
A. Waldronr Physics Letters B 433 (1998) 369–376
372
Fock space whose two-point function is DE Ž x E .. General Greens functions can be reconstructed via use of the Wick theorem. The time coordinate should play no preferred role ˆ in Euclidean space so the OS fields anticommute at all points x E and y E Ž x E and y E are now, of course, four dimensional.. Hence there is no time ordering in their Euclidean formulation and the Euclidean fields are expanded in terms of modes depending on Euclidean four-momenta. In any case there are no plane wave solutions to ŽyI E q m2 . c E s 0 in flat Euclidean space, so that, in the OS construction, Euclidean fields are off-shell. Hence their Euclidean fields have a form which is rather close to that of a five-dimensional Minkowski field at zero time. Explicitly the Euclidean canonical Dirac spinor of OS is given by
tion of hermitean conjugation is replaced by the composition of hermitean conjugation and a unitary involution Q defined as follows. For Bose fields f E Žt , x . Žwhich may be easily be treated in the OS approach without doubling, see w3x for details. Q acts simply as Euclidean time reversal,
cEŽ x.
x E† Ž x . d4 k
s
H
(Ž 2p . V
Bk ( Uk e i k x q D†k ( Vyk eyi k x 4 ,
4
Qf E Ž t , x . Qy1 s f E Ž yt , x . s f E Ž Q x . ,
Ž 22 .
where Q Žt , x . s Žyt , x .. This definition is motivated by the remark above that for continued Minkowski fields, one needed an additional Euclidean time reversal when comparing imaginary time evolution of the hermitean conjugated field with the hermitean conjugate of the imaginary time evolved field Žsee Ž18... For spinors however, the action of Q is more subtle. Define a field x E† Ž x ., the Euclidean analogue of the Minkowski field c Ž x,t ., by d4 k s
k
H
(Ž 2p . V
† eik x 4 . Bk† ( Wk†eyi k x q Dk ( Xyk
4
k
Ž 20 .
Ž 23 .
where V k ' Ž k 2 q m 2 .1r2 ,e i k x s e iŽ k 4tqkP x . and from now one we drop the subscript E on the Euclidean four-vectors k and x. The mode operators Bk and D†k act in a Euclidean Fock space with vacuum <0: E defined such that Bk <0: E s 0 s D k <0: E . The only non-vanishing anticommutation relations of the modes now give a four dimensional delta function,
Since we already doubled the number of spin polarizations Ž R,S s 1, . . . ,4., it would be a redoubling if the field x E† Ž x . were independent 5 of c E Ž x .. Rather, x E† is related to c E by both hermitean conjugation and the action of Q ,
Bk , Bk† 4 s d 4 Ž k y k . 1 s Dk , D†k 4 , X
x E† Ž t , x . ' Qy1c E† Ž yt , x . ig 0Q s Qy1c E† Ž Q x . ig 0Q .
Ž 21 .
Ž 24 .
where we again suppress polarization indices so that 1 denotes d R S. However, let us stress that the indices R and S no longer run over values 1,2, rather it is necessary to double the spin polarization degrees of freedom whereby R,S s 1, . . . ,4, and we denote Bk ( Uk ' Ý4Rs1 BkR UkR. The spinor wave functions Uk and Vyk do not satisfy any equations of motion. At this point the fields c E Ž x . in Ž20. and c Žyt , x . in Ž17. are totally unrelated, they act in different Fock spaces. The next task is to construct a conjugate momentum field to c E Ž x . Ži.e., the analogue of c † in the Minkowski case.. In the OS proposal, the answer is no longer w c E Ž x .x† Žalthough a key feature of the canonical formulation of our new Wick rotation is that this property will be retained., instead the opera-
The relation Ž24. together with the expansions Ž23. and Ž20. imply that Q has a more complicated action on the modes Žand in turn states in the Euclidean Fock space. such that
X
X
Qy1 BQ† k ( UQ†k ig 0Q s Bk† ( Wk† ; † Qy1 DQ k ( Vy† Q k ig 0Q s D k ( Xyk ,
Ž 25 .
where Q k s Žyk 4 , k .. Given the explicit forms Žsee w3x. of the spinor wave-functions Uk ,Vyk ,Wk and 5
Contrast this to an OS path integral approach for Dirac spinors in which the ‘‘doubling’’ of spinor degrees of freedom in Euclidean space is introduced by taking the field x E† to be independent of c E .
A. Waldronr Physics Letters B 433 (1998) 369–376
Xyk one can write down the action of Q on the mode operators Bk and Dk . For our purposes it is enough to note that the spinor wavefunctions satisfy spin polarization sums constructed such that one obtains the desired two-point function in Ž19., yiku Uk ( Wk† s
OS
qm
Vk
yiku † ;Vyk ( Xyk s
OS
ym
Vk
.
Ž 26 . Defining Q <0: E s <0: E one can then also calculate the action of Q on states in the Euclidean Fock space. The mode expansions Ž20. and Ž23. along with the spin polarization sums in Ž26. yield
c E Ž x . , x E† Ž y . 4 s 0 .
Ž 27 .
Furthermore, using Ž20., Ž21., Ž23. and Ž26. it is easy to verify the following equalities: DE Ž x y y . 'E ²0 < c E Ž x . x E† Ž y . <0: E s yE ²0 < x E† Ž y . c E Ž x . <0: E s D Ž yit q i s , x y y . , Ž 28 . Ž . Ž . Ž where De x y y in 28 agrees with D E x y y . in Ž19. so that the Euclidean two-point function without time ordering reproduces the continued, time ordered Minkowski two-point function. The final ingredient is the relation between states in the Euclidean Fock space, and those in the physical Minkowski Hilbert space. This is provided by the following mapping W : < X : E ™ < WX : M , Ž 29 . from an arbitrary state < X : E in the Euclidean Fock space to some state < WM : M in the Minkowski Hilbert space. We shall call this mapping the ‘‘OS-Wick map’’ and it is defined as follows. A general state in the Euclidean Fock space w3x may be represented as
the right. The OS-Wick map may now be defined by its action on the state < X : E in Ž30., W < X : E s < WX : M ' : c Ž yit 1 , x 1 . . . . c Ž yitmqn , x mqn . : <0: . Ž 31 . where, for brevity, we have suppressed the smearing by test functions f i Ž x i .. The fields c Žyit 1 , x 1 . and c Žyitmq n , x mq n . are precisely the continued Minkowski fields defined in Ž17. and Ž18., respectively, above. In w3x, the following central theorem is proven E
²Q X < Y : E sM ²WX < WY : M .
E
²Q X < X : E G 0 .
Ž 33 .
As yet we have made no mention of how dynamics are included in this proposal, but at this point we refer the reader to the original work of Osterwalder and Schrader w3x. Let us now give the generalization of the above construction to include our new Wick rotation.
3. The canonical formulation of the new Wick rotation In w3x, it is argued that the field x E† cannot be replaced by c E† since the two-point function, ²0 < c E Ž x . x E† Ž y . <0: E d4 k
The functions f i Ž x i . are some choice of test functions and it is convenient to normal order this expression as denoted by :: by which we mean all annihilation operators Bk and D k are to be pulled to
Ž 32 .
which states that inner products of states in the Euclidean Fock space are related to those in the Minkowski Hilbert space by the OS-Wick map and the unitary involution Q . The inner product in the Minkowski Hilbert space should be positive definite, whereby we immediately obtain the OS reflection positivity condition
E
Ž30.
373
s
H Ž 2p .
yiku OS q m 4
k E2 q m2
e i k E Ž xyy. E ,
Ž 34 .
is then inconsistent because only the left hand side is invariant under hermitean conjugation and the interchange of x and y. In light of our new Wick rotation, the remedy is obvious. One should replace ku OS by ku E s k Emg Em where the matrices g Em are defined in the introduction Žand were obtained via a
A. Waldronr Physics Letters B 433 (1998) 369–376
374
similarity transformation induced by the rotation of spinor indices. and put
x E† s yc E† g E5 .
Ž 35 .
In order to incorporate our rotation of spinor indices in the OS proposal, we replace the OS-Wick map W ˜ defined as follows: by a new OS-Wick map W,
˜ :M , W˜ : < X : ™ < WX
cEŽ x. s
Ž 36 .
d4 k
H(
Ž 2p .
4
Bk e i k x q Dk) eyi k x 4 ,
Ž 41 .
from which it follows that
where, in general, < X : E s :c E Ž x1 . . . .
four values. The necessity to reproduce the continued 3 q 1 dimensional Minkowski propagator, however, can be used to rule out directly replacing c E and x E† by their zero time, five dimensional counterparts. In this light, we consider the more general ansatz
c E†
Ž x nqm . Ž
yg E5
. : <0: E , Ž 37.
yc E† Ž x . g E5 d4 k
and
H(
Bk†g E5 eyi k x q Dki g E5 e i k x 4 ,
sy
˜ : M s :Sc Ž yit 1 , x 1 . . . . < WX
Ž 2p .
y1
: <0: M ,
4
Ž 38 .
Ž 42 .
with S s e . Hence, Euclidean Greens functions are related to their continued Minkowski Greens functions by an additional rotation of spinor indices. For example, for the free two-point function, we find
where we have replaced the combinations Uk ( Bk and D†k ( Vyk by the operator-valued four-spinors Bk and Dk) , respectively. We still require that Bk <0: E s 0 s Dk <0: E . By virtue of the rotation on spinor indices, the unitary involution Q is now defined on spinors in the same way as for bosons
=c Ž yitnqm , x nqm . S g 4 g 5p r4
E
²0 < c E Ž x . c E† Ž y . Ž yg E5 . <0: E d4 k s
H Ž 2p .
yiku E q m 4
k 2 q m2
e i kŽ xyy.
s Sy1 D Ž yit q i s , x y y . S .
Ž 39 .
Qc E Ž x . Qy1 s c E Ž Q x . ;Qc E† Ž x . Qy1 s c E† Ž Q x . , Ž 43 .
We must now construct the field c E , the Euclidean Fock space and the unitary involution Q . Let us briefly mention two unfruitful avenues before we give our solution. The first would be to require Wk† in Ž23. and Uk in Ž20. to satisfy
which in turn defines the action of Q on the spinor modes
Wk† s yUk†g E5
Clearly this action is involutive. Euclidean ultralocality,
Ž 40 .
Žalong with an analogous condition on Vyk and Xyk .. However, we must still reproduce the correct continued Minkowski two-point function which requires Uk ( Wk† s yiku E q m for which there is no solution when Ž40. holds 6 . A second, bolder proposal would be to notice that the expressions Ž20. and Ž23. are reminiscent of those of a five Ž5 s 1 q 4. dimensional Dirac spinor at time t 5 s 0, except that the polarization sums denoted by ‘‘(’’, should run over two, instead of
Q Bk†Qy1 s BQ† k , Q
Dki Qy1 s DQik
.
Ž 45 .
†
0 s c E Ž x . ,y c E Ž y . g E5 4 ,
Ž 46 .
is satisfied by requiring ) i , Dyk Bk , Bk† 4 s y Dyk 4. X
X
Ž 47 .
The correct two-point function as in Ž39. is ensured by the following anticommutation relations for the ‘‘spinor modes’’ y Bk , Bk†Xg E5 4 s
6 To see this, take a basis in which g E5 sdiagŽ1,1,y1,y1. and the other Dirac matrices are off-diagonal. Multiplying by yg E5 and tracing yields UkR ' 0Ž Rs1, . . . ,4..
Ž 44 .
E
yiku E q m
V k2
d 4 Ž k y kX . .
Ž 48 .
m .Žyg E5 . is hermitean with eigenThe matrix Ž yikuV kq 2 values "V k and may be diagonalized Žin a basis for
A. Waldronr Physics Letters B 433 (1998) 369–376
the Dirac matrices in which g E5 is diagonal. by defining 1 Bk s
(V
Uk B˜k , k
(2 Ž V y m . V k
The combination
k
†
s Ž Uy1 k . s Ž Uyk .
y1
½ B˜ , B˜ 5 s yg †X k
1 'V Uk is the analogue of the OS k
5 4 Ed
Ž k y kX . .
cEŽ x.
˜ (Ž 2p . V ½U B e k
4
k
ik x
q Uyk D˜ k) eyi k x ,
5
k
Ž 52 .
) i X , D˜ yk Ž k y k X . s y ½ D˜ yk 5, Ž 54 .
c E Ž x . , x E† Ž y . 4 ' y c E Ž x . , c E† Ž y . g E5 4 s 0 , Ž 55 . and possess the desired two-point function,
Ž 51 .
That the Euclidean Fock space now contains negative norm states causes no problems since we only require positiÕe norm in the Minkowski Hilbert space, which is assured since, by construction, our fields satisfy the same axioms as those of the OS proposal. Let us elaborate on this point since it constitutes one of the main observations of this note. The OS reflection positivity condition Ž33. is a statement about certain inner products in the Euclidean Fock space deriÕed from the positivity of inner products in the Minkowski Hilbert space. In the OS proposal, a Euclidean Fock space is employed in which the mode operators are doubled and all norms are positive but hermiticity, Ži.e., the relation between a spinor and its Dirac conjugate. are forsaken. Our observation is that one may trade positivity of norms in the Euclidean Fock space in lieu of hermiticity. This is possible since the Euclidean Fock space is doubled and indeed we find that exactly half of the Euclidean mode operators yield states having negative norm. To continue, we may ‘‘diagonalize’’ the Dk modes in a similar fashion so that the final result for our Euclidean fields reads
5 4 Ed
By construction these fields anticommute at all points in Euclidean space
E
d4 k
†X k
.
k
spinor wave function Uk . In this basis the mode relations now read
H
Ž 53 .
½ B˜ , B˜ 5 s yg Ž 50 .
s
x E† Ž x . s yc E† Ž x . g E5 s yQy1c E† Ž Q x . g E5 Q . The modes satisfy commutation relations
V k y m q iku E
k
and
Ž 49 .
where Uk s
375
²0 < c E Ž x . c E† Ž y . Ž yg E5 . <0: E d4 k s
H Ž 2p .
yiku E q m 4
k E2 q m 2
e i k E Ž xyy. E .
Ž 56 .
which is consistent with hermiticity. We have now reproduced the building blocks ŽŽ56., Ž54., Ž43. and Ž38.. of the OS construction and the rest of their proposal may now be inherited unaltered except for the replacement everywhere of the field x E† by yc E† g E5 and the extra rotation of spinor indices performed by the new OS-Wick map.
4. Conclusion In this note we have presented a generalization of the canonical work of Osterwalder and Schrader for Dirac spinors which fuses their approach with the Schwinger–Zumino w4,5x approach in which hermiticity is maintained although we still found it necessary to double the set of Euclidean fermionic mode operators, half of which now yield states having negative norm. Nonetheless, OS positivity is maintained, since, necessarily, Minkowski states have positive norm and the new proposal reproduces the same fundamental axioms as that of the original OS proposal. The essential new ingredient is our new Wick rotation for fermions under which spinor indices also rotate w1x. We would also suggest that our generalization adds to the formal simplicity of the OS construction.
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Finally, one may wonder what happens in the case of Majorana or Weyl spinors. In w1x we found that for Majorana and Weyl spinors in four dimensions that the requirement of hermiticity must be dropped. The extension of the work of OS to Majorana spinors was given by Nicolai w7x who noted that although there existed no consistent reality condition for Majorana spinors in four dimensional Euclidean space, one could nonetheless define a symplectic reality condition on the mode operators Bk and Dk following which the OS proposal may also be simply inherited. It is not a difficult matter to apply the generalization we have given above also to Nicolai’s work, and although one cannot regain hermiticity, the same formal algebraic simplifications as above occur. One may then even study N s 1 supersymmetric systems. As usual real bose fields undergo complex supersymmetry transformations in Euclidean space since the canonical supersymmetry charge Q no longer satisfies any reality condition. Such considerations may also be formulated in superspace Žsee also w8x..
Acknowledgements I am deeply indebted to Peter van Nieuwenhuizen for numerous detailed discussions and suggestions.
References w1x P. van Nieuwenhuizen, A. Waldron, Phys. Lett. B 389 Ž1996. 29; A Continuous Wick Rotation for Spinor Fields and Supersymmetry in Euclidean Space, in: Proceedings of Gauge Theories, Applied Supersymmetry and Quantum Gravity, London, World Scientific, 1996. w2x K. Osterwalder, R. Schrader, CMP 31 Ž1973. 83; CMP 42 Ž1975. 281. w3x K. Osterwalder, R. Schrader, Phys. Rev. Lett. 29 Ž1972. 1423; Helv. Phys. Acta 46 Ž1973. 277. w4x J. Schwinger, Phys. Rev. 115 Ž1959. 721. w5x B. Zumino, Phys. Lett. B 69 Ž1977. 369. w6x M. Blau, G. Thompson, Phys. Lett. B 415 Ž1997. 242. w7x H. Nicolai, Nucl. Phys. B 140 Ž1978. 284. w8x K. Schrader, in: Proceedings of the Conference on Advances in Dynamical Systems and Quantum Physics, Capri, World Scientific, 1993.