Supersymmetric Positivity and Supersymmetric Hilbert Space

Supersymmetric Positivity and

Supersymmetric Hilbert Space

FLORIN CONSTANTINESCUFachbereich Mathematik, Johann Wolfgang Goethe-Universitat Frankfurt,Robert-Mayer-Strasse 10, D 60054 Frankfurt am Main, Germany.

e-mail: [email protected]

(Received: 31 May 2002; revised: 4 July 2002)

Abstract. We introduce simple notions of positivity and Hilbert spaces of supersymmetricfunctions naturally suggested by the superspace formulation of supersymmetric quantum fieldtheory. Several applications are indicated.

Mathematical Subject Classifications (2000). 81Txx, 81T60, 46C05, 46Fxx.

Key words. Hilbert space, quantum field theory, superdistributions, supersymmetry.

1. Introduction

There have been many proposals for introducing positivity and a related Hilbert

space structure in SUSY theories; see, for instance, the recent publications [1, 2]

and references given there, as well as [3–5]. Whereas the approach in [2] is centered

on classical supersymmetry and quantization of supersymmetric solutions of classi-

cal field equations, the approach in [1] is motivated by the notion of physical obser-

vable as this appears in quantum mechanics and quantum field theory. The inner

products proposed in this Letter can be extended to Grassmann-valued inner pro-

ducts by omitting the integration over the Grassmann parameters and Hodge duali-

sation (for details, see [1]). It means that in the sense of Hilbert modules, these

Grassmann extended inner products, which closely follow physical work in super-

symmetric quantum field theory, are in fact examples of the general construction

in [1]. The physically interesting inner products can be obtained from the Grassmann

extended inner products by either taking the body ([1]) or by integrating over all

Grassmann parameters as this will be the case in this Letter.

Our definition is suggested by the unitarity of representations of the super-sym-

metric Poincare group ([3, 6]) when working in superspace. From this point of view,

our Letter is related to [5]. The difference is that in [5] the unitary representations

of the supersymmetric Poincare group are realized as unitary representations of a

large group algebra (or of some large set of functions on the group, which carries

the structure of a Hopf algebra), whereas our approach is entirely elementary and

starts from the beginning with a relatively small set of supersymmetric test functions

Letters in Mathematical Physics 62: 111–125, 2002. 111# 2002 Kluwer Academic Publishers. Printed in the Netherlands.

adjusted to the restricted problem under study. We restrict ourselves to the simplest

case of massive chiral/antichiral fields leaving other cases as for instance a Hilbert

space approach for supersymmetric gauge theories for future research. We introduce

here (chiral/antichiral) L2-spaces of supersymmetric functions similar to the usual

L2-spaces but without using measures on noncommutative variables. Needless to

say that they should be viable spaces. Contrary to the usual case in which L2-spaces

live for themselves, the supersymmetric L2-spaces are intimately connected to the

supersymmetry algebra. The same is true for test functions belonging to and the cor-

responding superdistributions embracing the L2-spaces used in this Letter. We pro-

vide some examples and applications concerning chiral/antichiral fields. In order to

describe (massive) vector fields, we need to enlarge our L2-spaces by another sector

associated to the so-called linear scalar (transverse) field ([7, 8]). This poses no special

problems besides ideas used in this Letter and, in order to preserve simplicity, we do

not include it explicitly.

Before starting, let us mention that in order to maintain the letter size of this paper

we give only few details of the proofs which are mainly of a computational nature.

2. Supersymmetric Positivity and Supersymmetric Hilbert Space

For the convenience of the reader, we use the notations and conventions of [9] with

only one change: our metric is ð1;�1;�1;�1Þ instead of ð�1; 1; 1; 1Þ. Consequently,

s0 and �s0 will be the identity instead of minus identity.

Consider supersymmetric chiral and antichiral (test) functions of the following

form which are defined by the conditions �Dj ¼ D �j ¼ 0 [9]:

jðx; y; �yÞ ¼ að yÞ þffiffiffi2

pycð yÞ þ y2fð yÞ

¼ aðxÞ þ iysm �y@maðxÞ � 14 y

2 �y2&aðxÞþ

þffiffiffi2

pycðxÞ þ

iffiffiffi2

p y2@mcðxÞsm �yþ y2fðxÞ; ð2:1Þ

�jðx; y; �yÞ ¼ �aðyþÞ þffiffiffi2

p�y �cðyþÞ þ �y2 �fð yþÞ

¼ �aðxÞ � iysm �y@m �aðxÞ � 14 y

2 �y2& �aðxÞþ

þffiffiffi2

p�y �cðxÞ �

iffiffiffi2

p �y2ysm@m �cðxÞ þ �y2 �fðxÞ; ð2:2Þ

with the components of y and yþ given by

ym ¼ xm þ iysm �y; yþm ¼ xm � iysm �y;

where the factorffiffiffi2

pis for convenience and we agree to take aðxÞ;cðxÞ; fðxÞ as func-

tions (or vector-valued functions in the fermionic case) in the Schwartz space SðR4Þ.

Note the signs in (2.1), (2.2) which are a consequence of our metric. Here D; �D are

the supersymmetric differential operators [9]:

112 FLORIN CONSTANTINESCU

Da ¼@

@yaþ isma�a �y

�a@m; ð2:3Þ

�D�a ¼ �@

@ �y�a � iyasma�a@m: ð2:4Þ

In terms of the variables y; y; �y or yþ; y; �y the operators D; �D have simpler expres-

sions [9]. We have D3 ¼ �D3 ¼ 0. An interesting property of the chiral/antichiral dif-

ferential operators is that D2j is antitichiral whereas �D2 �j is chiral. In forming this

expressions, the coefficients a;c and f undergo the action of the d’Alembertian. The

choice of test functions of the form (2.1), (2.2) is motivated by applications in physics

given in Section 3. The fellow mathematician with an aesthetic appeal for simplifica-

tion will find out that in fact the assumptions and the structure of this letter cannot

be simplified without getting almost the trivialities.

In Section 3 we will also consider functions (in coordinate and momentum space)

of type (2.1), (2.2) with coefficients satisfying ‘equations of motions’. They are alge-

braic or differential equations. We use the terminology ‘off-’ or ‘on-shell’ in a loose

sense meaning that the ‘equation of motions’ are fully or only partially satisfied. In

particular, we may restrict the coefficients in (2.1), (2.2) to the (forward) mass hyper-

boloid. This is natural and is reminiscent of the situation in relativistic field theory

(free fields).

Let us denote by SþSUSY and S�

SUSY the complex linear space of j’s and �j’s, respec-

tively. Eventually we will restrict j and �j to the (forward) mass hyperboloid (‘on-

shell’ test functions). If j 2 SþSUSY then �j 2 S�

SUSY and vice versa. We are going to

define positive (i.e. nonnegative) definite sesquilinear forms on S�SUSY � Sþ

SUSY. Let

us give a first example. We start by a simple computation which shows that for

j1;j2 2 SþSUSY we haveZ

d2yd2 �y �j1ðx; y; �yÞj2ðx0; y; �yÞ

¼ �f1ðxÞf2ðx0Þ � 1

4�a1ðxÞ&a2ðx

0Þ � 14& �a1ðxÞa2ðx

0Þþ

þ 12@m �a1ðxÞ@

ma2ðx0Þ �

i

2@m �c1ðxÞ �s

mc2ðx0Þ þ

i

2�c1ðxÞ �s

m@mc2ðx0Þ: ð2:5Þ

We will use several versions of (2.5). In particular, it follows from (2.5) that the

supersymmetric integralZd4xd4x0d2yd2 �y �jðx; y; �yÞð�iDþðx� x0ÞÞjðx0; y; �yÞ;

with

DþðxÞ ¼i

ð2pÞ3

Zyð p0Þdð p2 �m2Þe�ipxdp ð2:6Þ

is nonnegative. It vanishes if j ¼ 0 and it is strictly positive if j 6¼ 0 on shell. Here m

is the mass and DþðxÞ is the positive frequency Pauli–Jordan function. Up to the

SUPERSYMMETRIC POSITIVITY AND SUPERSYMMETRIC HILBERT SPACE 113

normalization constant, it is defined to be the positive frequency part of the Lorentz

invariant solution of the Klein–Gordon equation. The above integral can also be

written asZd4xd2yd2 �yd4x0d2y0d2 �y0 �jðx; y; �yÞKðx; y; �y;x0; y0; �y0Þjðx0; y0; �y0Þ;

where the kernel K is

Kðx; y; �y; x0; y0; �y0Þ ¼ dðy� y0Þdð�y� �y0Þð�iDþðx� x0ÞÞ:

This simple property will be the source of our supersymmetric positivity. Indeed, it

follows that for sþ0 defined as

sþ0 ðj1;j2Þ ¼

Zd4xd4x0d2yd2 �y �j1ðx; y; �yÞð�iDþðx� x0ÞÞj2ðx

0; y; �yÞ;

we have sþ0 ðj;jÞ > 0 for j 6¼ 0 on-shell in SþSUSY. Similarly,

s�0 ð �j1; �j2Þ ¼

Zd4xd4x0d2yd2 �yj1ðx; y; �yÞð�iDþðx� x0ÞÞ �j2ðx

0; y; �yÞ

is positive for �j1 ¼ �j2 6¼ 0 on-shell in S�SUSY. Note that for chiral functions j1;j2 we

have Zd4xd4x0d2yd2 �yj1ðx; y; �yÞð�iDþðx� x0ÞÞj2ðx

0; y; �yÞ ¼ 0

and a similar relation for antichiral ones.

All this enables us to define a positive definite inner product (scalar product) s0 on

complex linear combinations of chiral and antichiral functions, i.e. functions in SSUSY

of the form j ¼ j1 þ �j2;c ¼ c1 þ�c2 with j1;c1 2 Sþ

SUSY and �j2;�c2 2 S�

SUSY as

follows:

s0ðj;cÞ ¼Z

d4xd4x0d2yd2 �y �jðx; y; �yÞð�iDþðx� x0ÞÞcðx0; y; �yÞ:

Note the important conjugation property:

s0ðj;cÞ ¼ s0ð �j; �cÞ;

where the bar on the left-hand side is the usual complex conjugation whereas the bar

on the right-hand side includes the Grassmann conjugation. The scalar product s0can be also given using the kernel K:

s0ðj;cÞ ¼Z

d4xd2yd2 �yd4x0d2y0d2 �y0 �jðx; y; �yÞKðx; y; �y;x0; y0; �y0Þcðx0; y0; �y0Þ:

Other kernels will appear below.

By factorisation and completion of the zero vectors, we get from SSUSY the

Hilbert space L2SUSY of supersymmetric functions with L2-coefficients in front of


noncommutative variables. We denote by L2�SUSY, the chiral/antichiral part in L2

SUSY

such that we have the following orthogonal decomposition:

L2SUSY ¼ L2þ

SUSY � L2�SUSY:

The Hilbert spaces L2SUSY;L

2þSUSY;L

2�SUSY with scalar products s0; s

þ0 ; s

�0 respectively

are called L2-supersymmetric spaces.

In what follows we show that these Hilbert spaces are both natural and useful. The

orthogonal decomposition j ¼ j1 � �j2 in L2SUSY resembles similar decompositions

(polarizations) of usual L2-spaces used in order to describe chiral fiels, an appropri-

ate simple example being the construction of chiral fermionic Fock spaces in [10] in

the context of hyperfunctions.

Now we are going to introduce some (supersymmetric) kernels suggested by the

quantized chiral/antichiral free fields [9]. Indeed, if f ¼ fðx; y; �yÞ;fþ¼ fþ

ðx; y; �yÞare formal free chiral and antichiral fields, respectively, then we have the commutators

½fðx; y; �yÞ;fðx0; y0; �y0Þ� ¼ �imdðy� y0Þ exp½iðysm �y� y0sm �y0Þ@m�Dðx� x0Þ;

½fðx; y; �yÞ;fþðx0; y0; �y0Þ� ¼ �i exp½iðysm �yþ y0sm �y0 � 2ysm �y0Þ@m�Dðx� x0Þ;

½fþðx; y; �yÞ;fðx0; y0; �y0Þ� ¼ �i exp½�iðysm �yþ y0sm �y0 � 2y0sm �yÞ@m�Dðx� x0Þ;

½fþðx; y; �yÞ;fþ

ðx0; y0; �y0Þ� ¼ �imdð�y� �y0Þ exp½�iðysm �y� y0sm �y0Þ@m�Dðx� x0Þ;

where DðxÞ is the Pauli–Jordan function. By restricting this function to its positive

frequencies, we obtain the corresponding two-point functions which will be used

below. The commutators above can be easily obtained from the explicit multiplet

expansions of f and fþ [11] (see also [12]) being the (quantum) counterpart of the

corresponding propagators which appear in the physical literature [9].

We are going to define two new scalar products s1 and s2 on chiral/antichiral func-

tions. Both of them have applications which will be discussed in the next section.

Consider the kernel

Dþ22ðx; y; �y; x

0; y0; �y0Þ ¼ exp½iðysm �yþ y0sm �y0 � 2ysm �y0Þ@xm�ð�iDþðx� x0ÞÞ ð2:7Þ

obtained from ½f;fþ� above by replacing D by Dþ. When applied to a function

depending only on x and x0, it can be written in the equivalent form [8, 9]


0; y0; �y0Þ ¼�D2D2

16dðy� y0Þdð�y� �y0Þð�iDþðx� x0ÞÞ

¼D02 �D02

16dðy� y0Þdð�y� �y0Þð�iDþðx� x0ÞÞ; ð2:8Þ

where the operators D; �D act on the first variable [8] and D0; �D0 on the second

variable.

The kernel Dþ22ðx; y; �y; x

0; y0; �y0Þ defines an operator from SþSUSY to Sþ

SUSY i.e. for

every j in SþSUSY the integralZ

d4x0d2y0d2 �y0Dþ22ðx; y; �y; x

0; y0; �y0Þjðx0; y0; �y0Þ ð2:9Þ


belongs to SþSUSY. The proof of this assertion is by supersymmetric partial integra-

tion [8].

Now we state our next result. It says that

sþ1 ðj1;j2Þ ¼

Zd4xd2yd2 �yd4x0d2y0d2 �y0 �j1ðx; y; �yÞD

þ22ðx; y; �y; x

0; y0; �y0Þj2ðx0; y0; �y0Þ

ð2:10Þ

is positive for j1 ¼ j2 6¼ 0 on-shell in SþSUSY.

The proof follows by computation preferably in Fourier space using the positivity

of p2 and sp on the forward mass hyperboloid (in the case of sp both determinant

and trace are positive).

A similar construction works when the kernel Dþ22ðx; y; �y; x

0; y0; �y0Þ is replaced by its

conjugate resulting from ½fþ;f�:

Dþ11 ¼ exp½�iðysm �yþ y0sm �y0 � 2y0sm �yÞ@x

m�ð�iDþðx� x0 ÞÞ; ð2:11Þ

which can be also written when applied to a function of only x; x0 as

Dþ11ðx; y; �y; x; y

0; �y0Þ ¼D2 �D2


¼�D02D02

16dðy� y0Þdð�y� �y0Þð�iDþðx� x0ÞÞ: ð2:12Þ

As above, the operators D; �D act on the first variable and the operators D0; �D0 on the

second variable. In this way we obtain scalar products sþ1 ; s�1 on the space of chiral

and antichiral functions, respectively. We will now consider the (complex) space

SSUSY of linear combinations of chiral and antichiral functions. Taking into account

the relationsZd4xd2yd2 �yd4x0d2y0d2 �y0j1ðx; y; �yÞD

þ22ðx; y; �y; x

0; y0; �y0Þj2ðx0; y0; �y0Þ ¼ 0

and Zd4xd2yd2 �yd4x0d2y0d2 �y0 �j1ðx; y; �yÞD

þ22ðx; y; �y; x

0; y0; �y0Þ �j2ðx; y; �yÞ ¼ 0

for j1;j2 2 SþSUSY together with other two symmetric relations involving Dþ

11, we can

define the scalar product

s1ðj;cÞ ¼Z

d4xd2yd2 �yd4x0d2y0d2 �y0 �jðx; y; �yÞðDþ11ðx; y; �y; x

0; y0; �y0Þþ

þDþ22ðx; y; �y; x

0; y0; �y0Þcðx0; y0; �y0Þ; ð2:13Þ

where j and c are both linear combinations of chiral and antichiral supersymmetric

functions (i.e. are elements of SSUSY). We use the fact that �ð1=16&Þ �D2D2 projects

on chiral, whereas �ð1=16&ÞD2 �D2 on antichiral functions [9] and in L2SUSY we


have �ð1=16&ÞðD2 �D2 þ �D2D2Þ ¼ 1 to find out that on shell the scalar product s1 is

proportional to s0. In passing, we remark that within our framework the (formal)

projections

P1 ¼ �1

16&D2 �D2 and P2 ¼ �

1

16&�D2D2

become Hilbert space bona-fide orthogonal projection operators. On shell, i.e. on the

mass hyperboloid, we can replace & by �m2. Note again the conjugation property

of the scalar product involving the Grassmann beyond the complex conjugation.

Before passing to the even more interesting scalar product s2, we remark that our

framework up to now can be enlarged to include a third sector in the Hilbert space

direct sum decomposition

L2SUSY ¼ L2þ

SUSY � L2�SUSY � L2T

SUSY

of the L2-supersymmetric space. The corresponding scalar product can be generated

as above by the kernel PTKðx; y; �y; x0; y0; �y0Þ, where PT is the linear transvesal projec-

tion [7–9]. It can be shown that this is the most general (scalar) L2-space of super-

symmetric functions. It is the right framework to start a Hilbert space study of

supersymmetric gauge theories. We do not elaborate on this point and continue

our study of the chiral/antichiral sectors.

We define our third scalar product s2. Considerations regarding the so-called ‘off-

shell’ formulation [9] suggest the introduction of the linear space S]SUSY with ele-

ments j] ¼ ðj1

j2Þ, where j1;j2 2 SSUSY. We introduce also the spaces S

]�SUSY. The ele-

ment j] belongs to S]þSUSY if j1 2 Sþ

SUSY;j2 2 S�SUSY; the definition of S

]�SUSY is

similar. Let us consider the matrix kernel

D]þðx; y; �y; x0; y0; �y0Þ ¼Dþ

22ðx; y; �y;x0; y0; �y0Þ Dþ

12ðx; y; �y; x0; y0; �y0Þ

Dþ21ðx; y; �y;x

0; y0; �y0Þ Dþ11ðx; y; �y; x

0; y0; �y0Þ

!; ð2:14Þ

where Dþ11 and Dþ

22 were already introduced in [9, 11] and


0; y0; �y0Þ

¼ mdð�y� �y0Þ exp½�iðysm �y� y0sm �y0Þ@xm�ð�iDþðx� x0ÞÞ; ð2:15Þ


0; y0; �y0Þ

¼ mdðy� y0Þ exp½iðysm �y� y0sm �y0Þ@xm�ð�iDþðx� x0ÞÞ ð2:16Þ

are read-off from ½fþ;fþ� and ½f;f�, respectively. We have equivalent formulas [8, 9]


0; y; �y0Þ ¼ mD2


¼ mD02




0; y; �y0Þ ¼ m�D2


¼ m�D02


where the operators D; �D act on the first variable and D0; �D0 on the second variable

(as above kernels are suppose to multiply under the supersymmetric integral func-

tions of variables x; x0). Note that Dþ22 is conjugate to Dþ

11 and Dþ21 to Dþ

12 which is

a consequence of the relation

�iDþð�xÞ ¼ �iDþðxÞ

for the positive frequency Pauli–Jordan function.

The kernels Dþ12ðx; y; �y; x

0; y0; �y0Þ and Dþ21ðx; y; �y; x

0; y0; �y0Þ define operators from

SþSUSY to S�

SUSY and from S�SUSY to Sþ

SUSY respectively.

The matrix kernel D]þðx; y; �y; x0; y0; �y0Þ defines an operator on S]þSUSY. It also

defines a (positive) scalar product

s2ðj];c]Þ ¼

Zd4xd2yd2 �yd4x0d2y0d2 �y0 �j]ðx; y; �yÞD]þðx; y; �y; x0; y0; �y0Þc]

ðx0; y0; �y0Þ

on S]þSUSY.

As in the case of s0 and s1 we introduce the scalar product s2 on S]�SUSY given by

s2ðj];c]Þ ¼

Zd4xd2yd2 �yd4x0d2y0d2 �y0 �j]ðx;y; �yÞED]þðx;y; �y;x0;y0; �y0ÞEc]

ðx0;y0; �y0Þ;

and finally the scalar product s2 on S]SUSY given by

s2ðj];c]Þ ¼

Zd4xd2yd2 �yd4x0d2y0d2 �y0 �j]ðx; y; �yÞDþðx; y; �y; x0; y0; �y0Þc]

ðx0; y0; �y0Þ;

where Dþ is the matrix kernel

Dþðx; y; �y; x0; y0; �y0Þ ¼ D]þðx; y; �y;x0; y0; �y0Þ þ E �D]þðx; y; �y; x0; y0; �y0ÞE

and E ¼�

0 1

1 0

�. The proof is again by computation. Explicitly, Dþ is given by

116 ð

�D2D2 þD2 �D2Þ m4 ðD

2 þ �D2Þm4 ð

�D2 þD2Þ 116 ðD

2 �D2 þ �D2D2Þ

� �dðy� y0Þdð�y� �y0Þð�iDþðx� x0ÞÞ:

Using the projection relation above, we can bring the matrix kernel Dþ in the chiral/

antichiral sector to the simplified form

Dþ ¼ m2 1 14m ðD

2 þ �D2Þ1

4m ð�D2 þD2Þ 1

� �dðy� y0Þdð�y� �y0Þð�iDþðx� x0ÞÞ:

But now even on the mass hyperboloid zero vectors may appear which are given by

the ‘equations of motions’ for the test functions presented in the next section. By


factorization and completion we get a Hilbert space L2]SUSY. There is again an ortho-

gonal decomposition

L2]SUSY ¼ L

2]þSUSY � L

2]�SUSY

associated separately with the kernels D]þ; ED]þE which is analogous to the previous

one but we do not describe it explicitly here.

3. Applications, Conclusions and Remarks

We have introduced supersymmetric positivity and supersymmetric L2-spaces with

respect to the integration over superspace. Hilbert scalar products defined above

generate Hilbert norms (seminorms) and enable us to define superdistributions.

We now indicate some applications. One could be the axiomatic framework [13]

for (interacting) massive chiral/antichiral fields which can be constructed as usual

using the locally convex tensor algebra over the ‘off-shell’ space S]SUSY. The positivity

is a (nonlinear), condition on the n-point functions and the fields can be reconstruc-

ted from the n-point functions. It would be interesting to explore the consequences of

the ‘superanalyticity’ generated in the configuration space by the positivity condi-

tion. We do not elaborate on this point. Instead, we give several (unitary equivalent)

genuine supersymmetric Fock space representations of the free chiral/antichiral

quantum field fðx; y; �yÞ= �fðx; y; �yÞ which can be looked at as nontrivial (i.e., no tensor

product) superspace transcriptions of the Fock space representations of the compo-

nent fields. Our symmetric supersymmetric (!) Fock spaces are constructed with the

help of scalar products s2 and s1. These representations look similar to the Fock

space representations of the scalar field (see, for instance, [13]) now taken over

our supersymmetric test functions in L2]SUSY or L2

SUSY. Here it is interesting to remark

that the corresponding Fock spaces are symmetric and do not have the choice

between symmetry and antisymmetry as in the usual case. Let us denote

f]¼

�ffþ

�. It is denoted by F in [9]. The arbitrary smearing test functions for f]

are denoted by

j] ¼jc

� �2 L

2]SUSY

such that in the sense of (super)distributions

fðjÞ ¼Z

d4xd2yd2 �yfðx; y; �yÞjðx; y; �yÞ

and a similar relation for fþðcÞ. Note that in order to obtain chiral/antichiral fields

common in physics we have to smear with canonical test functions, i.e. to smear the

chiral field with an antichiral test function (i.e. to form fð �jÞ for �j 2 L2�SUSY ) and

other way around. Indeed fðjÞ ¼ 0 for j 2 L2þSUSY. In particular, one obtains

fðx; y; �yÞ when smearing fðx0; y0; �y0Þ by the supersymmetric delta-function


d4ððx� x0Þd2

ðy� y0Þd2ð�y� �y0Þ (strictly speaking by the projection of it on the antic-

hiral sector). Note that the delta-function d4ðx� x0Þd2

ðy� y0Þd2ð�y� �y0Þ and its

chiral/antichiral projections are supersymmetric invariant. Let us restrict j] to

L2]�SUSY (canonical test functions). We have the following Fock space representation

on canonical test funtions:

ðf]ðj]Þc]

Þnðo1; . . . ;onÞ

¼ffiffiffip

p½n�

12

Xni¼1

j]ðoiÞc]n�1ðo1; . . . ; oi; . . . ;onÞþ

þ ðnþ 1Þ12h �j]ðo�Þ;c]

nþ1ðo;o1; . . . ;onÞi�; j] 2 L]�SUSY ð3:1Þ

where on the right-hand side the inner product h:i (to be given below) is written in the

p-variable Fourier space, oi ¼ ðpi; yi; �yiÞ , in o� it is understood that p is replaced by

�p and c]n is the n-particle component of the supersymmetric Fock space vector c].

As usual, the hat over the argument means omission. The vacuum O is identified

through j]O ¼ j].

The inner product h:; :i in (3.1) is not positive definite because c] is not Hermitean.

It is related to the scalar product s2 being given by the kernel

Dþðx; y; �y;x0; y0; �y0Þ

¼ EDþðx; y; �y; x0; y0; �y0Þ ¼ Dþðx; y; �y; x0; y0; �y0ÞE

¼ðDþ

12 þDþ21Þðx; y; �y; x

0; y0; �y0Þ ðDþ11 þDþ

22Þðx; y; �y;x0; y0; �y0Þ

ðDþ11 þDþ

22Þðx; y; �y; x0; y0; �y0Þ ðDþ

12 þDþ21Þðx; y; �y;x

0; y0; �y0Þ

!

as

hj];c]i ¼

Zd4xd2yd2 �yd4x0d2y0d2 �y0 �j]Tðx; y; �yÞDþ

ðx; y; �y;x0; y0; �y0Þc]ðx0; y0; �y0Þ;

where T means transposition. As opposed to the (positive) s2 the indefinite h:; :i con-

tains the twist E. The inner product h:; :i differs from the (positive definite) scalar pro-

duct ð:; :Þ � s2ð:; :Þ by a twist given by the matrix E.With the help of this kernel, the reader can write down the causal commutation

relations for our chiral/antichiral quantum field f] recovering the field commutators

in Section 2. Note that, in order to obtain the usual results, we have restricted (as

mentioned above) the general smearing function j] 2 L2]SUSY in f]

ðj]Þ to j] ¼ ð�jcÞ

with �j 2 L2�SUSY or c 2 L2þ

SUSY; i.e. j] 2 L2]�SUSY. Otherwise f]

ðj]Þ ¼ 0.

It is useful to extend the operator f] from L2]�SUSY to L

2]SUSY by defining for

j] 2 L2]SUSY:

ðf]ðj]Þc]


¼ffiffiffip

p hn�

12

Xni¼1

P2j]ðoiÞc]n�1ðo1; . . . ; oi; . . . ;onÞþ

þ ðnþ 1Þ12hP2j]ðo�Þ;c]

nþ1ðo;o1; . . . ;onÞi

i; ð3:2Þ


where P2 is the projection operator from L2]SUSY to L

2]þSUSY which is given by

P2 ¼ �1

16&D2 �D2 0

0 �D2D2

� �: ð3:3Þ

Now consider the formal adjoint

Fþ � f]þ¼

fþ

f

� �and for j] 2 L

2]SUSY let us define the Fock space operator

ðf]þðj]Þc]


¼ffiffiffip

p½n�

12

Xni¼1

P2Ej]ðoiÞc]n�1ðo1; . . . ; oi; . . . ;onÞþ

þ ðnþ 1Þ12hP2Ej]ðo�Þ;c]

nþ1ðo;o1; . . . ;onÞi�; ð3:4Þ

where P2E can be also written as EP1 and

P1 ¼ �1

16&�D2D2 00 D2 �D2

� �: ð3:5Þ

If we write f]ðj]Þ

þ for the operator adjoint of f]ðj]Þ, we have the relation

f]ðj]Þ

þ¼ f]þ

ð �j]Þ together with the rigorous (smeared) version of the commutation

relations. We leave the computations as an exercise for the reader. They are similar

to the corresponding computations in the case of the neutral scalar field [13]. The

only new point here is the fact that the inner product h:; :i in the field representation

does not coincide with the Hilbert (Fock) scalar product ð:; :Þ � s2ð:; :Þ as a conse-

quence of the non-Hermiticity of f]ðj]Þ. ð:; :Þ and h:; :i are related by the twist E.

Let us come back to the representations (3.2). We can eliminate (factorise) the zero

vectors by projecting (3.2) on the orthoginal complement of the subspace of the

‘equations of motion’ for the test functions j] ¼

�jc

�:

jþ1

4mðD2 þ �D2Þc ¼ 0; cþ

1

4mð �D2 þD2Þj ¼ 0:

They give the zero vectors of D which were found to be nonnegative definite. The

‘equations of motion’ for the chiral/antichiral fields are [9]:

mf ¼ 14�D2fþ; mfþ

¼ 14D

2f:

Within our framework in which we do not use Lagrangians, they are taken as defi-

nitions. The different signs in the equations of motion for test functions and fields

will make it possible to find a simple one-component functional Fock space represen-

tation from the factorized two-component representation (3.1). Indeed, it is obvious

that the projections on the equations of motion for test functions and fields are

orthogonal. It means that instead of factorising zero vectors, we can formally project

on the equations of motions for fields but applied to test functions. The result of this

projection is described below under (3.6) and (3.7).

The field equations of motion imply in particular the Klein–Gordon and Dirac

equations for the scalar and Majorana components of f and fþ, respectively. Note


that the ‘equations of motion’ for the test functions are symmetrised as opposed to

the ‘physical’ equations of motions for the fields. In fact this symmetrisation appears

throughout this Letter at the level of test functions and enables us to treat the chiral

and the antichiral sectors on the same footing. In the process of computing two-

point funtions and commutators, orthogonality selects the right contributions from

the symmetrised kernels. This is typical for chiral fields.

We obtain by the projection described above

ðfðjÞcÞnðo1; . . . ;onÞ

¼ffiffiffip

pn�

12

Xni¼1

jðoiÞcn�1ðo1; . . . ; oi; . . . ;onÞþ

"

þ ðnþ 1Þ12

Zd4pd2yd2 �yd2y0d2 �y0jðo�ÞðDþ

12 þDþ21Þ�

�ð p; y; �y; y0; �y0Þcnþ1ðo;o1; . . . ;onÞ

#; ð3:6Þ

where Dþ21ð p; y; �y; y

0; �y0Þ is the Fourier transform of

Dþ21ðx; y; �y; x; y

0; �y0Þ ¼ m�D2

4dðy� y0Þdð�y� �y0Þð�iDþðxÞÞ

and the rest of notations are self-explanatory. The Fock space vacuum is the (super-

symmetric) function 1.

Analogously we formally get

fþðjÞ ¼

1

4mðD2fÞðjÞ ¼

1

4mððD2 þ �D2ÞfðjÞ ¼

1

4mfððD2 þ �D2ÞðjÞÞ;

obtaining

ðfþðjÞcÞnðo1; . . . ;onÞ

¼ffiffiffip

p 1

4mn�

12

Xni¼1

ðD2 þ �D2ÞjðoiÞcn�1ðo1; . . . ; oi; . . . ;onÞþ

"

þm2ðnþ 1Þ12

Zd4pd2yd2 �yjðo�Þcnþ1ðo;o1; . . . ;onÞ

#;

ð3:7Þ

where we have used supersymmetric partial integration [8].

The rigorous proof of (3.7) can be given by computing the adjoint of fðjÞ from

(3.6).

The plane wave decompositions of the chiral/antichiral fields are given by

fðx; y; �yÞ ¼1

ð2pÞ32

Z½að �p; y; �yÞe�ipx þ bþð �p; y; �yÞeipx�

d3 �pffiffiffiffiffiffiffi2p0

p ; ð3:8Þ

fþðx; y; �yÞ ¼

1

ð2pÞ32

Z½bð �p; y; �yÞe�ipx þ aþð �p; y; �yÞeipx�


p ; ð3:9Þ

where p ¼ ð p0; �pÞ is the Fourier variable and p0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�p2 þm2

p.


We have

að �p; y; �yÞ ¼ �i

Zd3xeipx@0

$

fðx; y; �yÞ;

bð �p; y; �yÞ ¼ �i

Zd3xeipx@0

$�fðx; y; �yÞ;

aþð �p; y; �yÞ ¼ �i

Zd3xe�ipx@0

$�fðx; y; �yÞ;

bþð �p; y; �yÞ ¼ �i

Zd3xe�ipx@0

$

fðx; y; �yÞ;

where, as usual, u@$

v ¼ uð@vÞ � ð@uÞv:

The commutation relations of the operators aþ; bþ and a; b in (3.8), (3.9) (which

we could call in an improper way supersymmetric ‘creation’ and ‘annihilation’ opera-

tors) are

½að �p; y; �yÞ; bþð �p0; y; �y0Þ� ¼ m�D2 þD2

4dð �p� �p0Þdðy� y0Þdð�y� �y0Þ; ð3:10Þ

½að �p; y; �yÞ; aþð �p0; y; �y0Þ� ¼�D2D2 þD2 �D2


½bð �p; y; �yÞ; bþð �p0; y; �y0Þ� ¼D2 �D2 þ �D2D2

16dð �p� �p0Þdðy� y0Þdð�y� �y0Þ ð3:12Þ

½bð �p; y; �yÞ; aþð �p0; y; �y0Þ� ¼ mD2 þ �D2


where now the D-operators are taken in p-space

Da ¼@

@yaþ isma�a �y

�apm; �D�a ¼ �@

@ �y�a � iyasma�apm

the rest of the commutators being zero. In (3.11) and (3.12), we can substitute

D2 �D2 þ �D2D2 ¼ 16p2 ¼ 16m2 because these relations are understood as being

smeared by on-shell chiral/antichiral supersymmetric functions in L2SUSY. The

‘creation’ and ‘annihilation’ operators (3.10)–(3.13) are unitarily represented on sym-

metric Fock spaces constructed by means of on-shell supersymmetric test functions

of the form (2.1), (2.2) by the above field representation (3.2). In the case projected

on the ‘equations of motion’, the plane-wave decomposition and the commutation

relations for the ‘creation’ and ‘annihilation’ operators are especially simple because

the above operators become dependent:

bð �p; y; �yÞ ¼D2 þ �D2

4mað �p; y; �yÞ: ð3:14Þ

Then

fðx; y; �yÞ ¼1

ð2pÞ32

Z½að �p; y; �yÞe�ipx þ

D2 þ �D2

4maþð �p; y; �yÞeipx�


p ð3:15Þ


A bunch of relations between the supersymmetric ‘creation’ and ‘annihilation’ opera-

tors and the creation and annihilation operators of the component fields can be given

but they do not seem to be very enlightening.

Having represented the free chiral/antichiral fields as bona-fide operator-valued

superdistributions, we can define as usual Wick products as operators in our Fock

spaces. In particular, it is easy to realise that, for pure chiral or pure antichiral mono-

mials, the Wick prescription is superfluous. Finally, we can define products of Wick

powers and prove the so-called zero theorem of Epstein and Glaser [14]. This is

the starting point of a renormalization method invented by Epstein and Glaser at

the end of the sixties [14]. By the methods of this paper we can formulate the

Epstein–Glaser method for the Wess–Zumino model in a fully supersymmetric

way. Besides supersymmetric invariance our supersymmetric positivity and super-

symmetric Hilbert space provide the important property of super unitarity (i.e. uni-

tarity in the supersymmetric Hilbert space) for the formal series expansion of the

S-matrix operator. Already the inspection of the distribution-theoretic singularity

problem based on our particular test functions either as splitting [14] or as extension

(see, for instance, [15]) shows that scaling x ! lx; y ! l12y; �y ! l

12 �y is powerful

enough in order to control renormalizability of the Wess–Zumino model including

the nonrenormalization theorem (for a treatment which can be made rigorous by

the present methods, see [16]).

References

1. Rudolph, O.: Comm. Math. Phys. 214 (2000), 449.2. Deligne, P. et al. (eds), Quantum Fields and Strings: A Course for Mathematicians, Amer.

Math. Soc., Providence, 1999.

3. Ruhl, W. and Yunn, B. C.: Fortschr. Phys. 23 (1975), 431; 23 (1975), 451.4. Nagamachi, S. and Kobayashi, Y.: Axioms of supersymmetric quantum fields, Unpub-

lished preprint

5. Osterwalder, K.: Supersymmetric quantum field theory, In: V. Rivasseau (ed), Results inField Theory, Statistical Mechanics and Condensed Matter Physics, Lecture Notes in Phys.446, Springer, New York, 1995, 117.

6. Ferrara, S., Savoy, C. A. and Zumino, B.: Phys. Lett. B 100 (1981), 393.7. West, P.: Introduction to Supersymmetry and Supergravity, 2nd edn, World Scientific,

Singapore, 1990.8. Srivastava, P. P.: Supersymmetry, Superfields and Supergravity: An Introduction, IOP Pub-

lishing, Adam Hilger, Bristol, 1986.9. Wess, J. and Bagger, J.: Supersymmetry and Supergravity, 2nd edn, Princeton Univ. Press,

1992.

10. Constantinescu, F. and Scharf, G.: Lett. Math. Phys. 52 (2000), 113.11. Constantinescu, F., Gut, M. and Scharf, G.: Quantized superfields, Ann. Phys. ðLeipzigÞ

11 (2002), 335.

12. Lopuszanski, J.: An Introduction to Symmetry and Supersymmetry in Quantum Field The-ory, World Scientific, Singapore, 1991.

13. Jost, R.: The General Theory of Quantized Fields, Amer. Math. Soc., Providence, 1965.

14. Epstein, H. and Glaser, V.: Ann. Inst. H. Poincare A 19 (1973), 211.


15. Prange, D.: Kausale Storungstheorie und differentielle Renormierung, Diplomarbeit, II.

Institut fur Theoretische Physik, Universitat Hamburg, 1997.16. Gates, S. T., Jr., Grisaru, M. T., Rocek, M. and Siegel, W.: Superspace, Benjamin,

New York, 1983.


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Supersymmetric Positivity and Supersymmetric Hilbert Space