International Mathematical Forum, Vol. 14, 2019, no. 4, 169 - 179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2019.9728

Group Action on Quantum Field Theory

Ahmad M. Alghamdi and Roa M. Makki

Department of Mathematical Sciences Faculty of Applied Sciences Umm Alqura University P.O. Box 14035, Makkah 21955, Saudi Arabia

This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd.

Abstract The main aim of this paper is to develop some tools for understand- ing and explaining the concept of group action on a space and group action on algebra by using subgroups with ﬁnite index and both re- striction and transfer maps. We extend this concept on quantum ﬁeld theory.

0 Introduction

Group action is a very important tool in mathematics. Namely, if a certain group acts on a certain object, then many results and phenomena can be resolved. Let G be a group, the object can be a set and then we have the notion of a group acts on a set see [7]. Such object can be a vector space which results the so called G-space or G-module see for instance [6]. In some cases, the object can be a group and then we have the notion of group acts on a group see [3]. The topic of a group acting on an algebra can be seen in [5, 9]. This approaches open many gates in the research area called the G-algebras over a ﬁeld. If we have an action from the group G to an object, then there is a ﬁxed point under this action. This is the key idea in the theory of group action. In fact, by restriction, each subgroup of G acts on the same object too. Therefore, that subgroup has also ﬁxed points. This theory goes to back the work of Cauchy Frobenius Burnside see [10]. That is the counting lemma which has many application in real life. However, the systematic method which had been initiated by Green in [4] is a very signiﬁcant approach to unify the study of group action on certain objects. 170 Ahmad M. Alghamdi and Roa M. Makki

That paper gives a fundamental study of the group action on ﬁnite dimensional algebras over a ﬁeld. But such study was for ﬁnite groups. Our main object in this work is to deal with inﬁnite groups and then make restriction for the subgroups of those inﬁnite groups with certain condition. To further study these actions, we use the restriction and transfer maps as main tools. We assume all the time that the index of any subgroup H of the group G is ﬁnite even if G was inﬁnite to insure that the sum of certain series is convergent. As a notion of the index, we may write that [ G : H] . We write V to denote a vector space over a ﬁeld F . We organize the paper as follows. Section 1 for some deﬁnitions and prelim- inaries. Section 2 is devoted to the general construction of the related quotient space and a subsection for the action of group on algebra. Section 3 is for the new notion which we initiated, namely group action on quantum ﬁeld theory.

1 Deﬁnitions and Preliminaries

In this section, we recall the basic deﬁnitions which are related to our work and that we shall use frequently. We start by deﬁning what is a G-space.

Deﬁnition 1.1 Let G be a group that acts on a vector space V over a vector ﬁeld F . For all g ∈ G and v ∈ V , there exist a unique element say vg ∈ V such g that there exists a unique element say v.g or v in V where (v1+v2)g = v1g+v2g and v(g1g2) = (vg1)g2 for all v1, v2 ∈ V and g1, g2 ∈ G.

If G is a group that acts on a vector space V over a ﬁeld F , then we deﬁne the ﬁxed point, which is still a subspace, as follows:

Deﬁnition 1.2 Let G be a group and V a vector space over a ﬁeld F . Assume that G acts on V , then we deﬁne a ﬁxed point of such action by

InvG(V ) = {v ∈ V : v = vg, ∀g ∈ G}.

The invariant set deﬁned in Deﬁnition 1.2 is a G-subspace.

Let H be a subgroup of G, then the ﬁxed point of G acting on each H can be denoted by InvH (V ). Lemma 1.3 Let G be a group that acts on a subgroup H of G, then

Hg = {g−1hg , for h ∈ H} is a subgroup of G and InvG(V ) ⊆ InvH (V ). Group action on quantum ﬁeld theory 171

g −1 −1 Proof: H is a nonempty set since, for 1G ∈ G we have g 1G g = g g = g −1 −1 1G ∈ H. Now take h1, h2 ∈ H such that h1 = g xg and h2 = g yg for −1 −1 −1 −1 −1 x, y ∈ H, then h1h2 = g xgg yg = g (xy)g ∈ g Hg.

Lemma 1.4 Let G be a group that acts on V , then for all g ∈ G

InvHg (V ) = InvH (V ).g

Proof: One inclusion is clear and the other inclusion follows from the deﬁ- nition of the action.

We mentioned that our work will be an action of inﬁnite group on certain objects in QFT. That inﬁnite group is mainly the automorphism group of that object or a subgroup of it. Let us remind the reader that the general linear group which is the automorphism group of a certain vector space contains all groups under consideration. That group can be equipped with certain length and norm which can be used to look at it as a topological group. We refer the reader for more information to see the book by Baker [1].

We deﬁne the restriction and transfer maps as follows:

G Deﬁnition 1.5 A restriction map is deﬁned as RH : InvG(V ) → InvH (V ) G such that for a ∈ InvG(V ), we have RH (a) = a. For a group G and a subgroup H of that G, one can deﬁne a relation on G such that for all x, y ∈ G, x is related to y if and only if Hx = Hy. It follows that such relation is an equivalence relation. This equivalence relation generates equivalence classes which is called co-sets. We shall assume in this work that all subgroups we are dealing with have ﬁnite index. We ﬁx T to be an orbit of this relation which is called a co-set representative or a transversal set.

Deﬁnition 1.6 Let G be a group and H a subgroup of G such that the index [G : H] = n < ∞. Fix a transversal set T = {g1, g2, ..., gn}. A transfer map is G deﬁned as TH : InvH (V ) → InvG(V ), such that for a ∈ InvH (V ), we have:

n G X TH (a) = ag1 + ag2 + ... + agn = agi. i=1

We remark that the transfer map has more than one name in the literature such as relative trace map as well as Gasch¨utzoperator (see [5] page 36). In the language of representation theory, one can consider the relationship with 172 Ahmad M. Alghamdi and Roa M. Makki

G the induction of an H-space to be the G-space IndH (V ) as a consequence of the transfer map. It is clear that the transfer map is a linear map and if a is ﬁxed under the G H action, then its image is ﬁxed under G action. For TH is a linear map, we G G G G G see that TH (a + b) = TH (a) + TH (b) and TH (ka) = kTH (a) for all k ∈ F . Also

G TH (a) ∈ InvG(V ), ∀a ∈ InvH (V ). We have more properties such as:

G G Lemma 1.7 TH (V ) = THg (V ), for g ∈ G. Proof: Clear.

We have the following tools:

Deﬁnition 1.8 G G TH ◦ RH : InvG(V ) → InvG(V ), G G RH ◦ TH : InvH (V ) → InvH (V ).

G G By this deﬁnition, we see that the composite map: TH ◦ RH can be seen as an G G endomorphism of the space InvG(V ). Similarly, the map RH ◦ TH can be seen as an endomorphism of the space InvH (V ).

Deﬁnition 1.9 Let H 6 K 6 G, then InvG(V ) ⊆ InvK (V ) ⊆ InvH (V ).

G G K We note that, if H 6 K 6 G, then TH = TK ◦ TH . This property is called the transitivity of the transfer map. For notstion use, we have the following remark.

g Remark 1.10 If G is a group and H 6 K for some g ∈ G, then we write H 6G K.

G G Corollary 1.11 If H 6G K, then TH (V ) ⊆ TK (V ). The main idea in this paper is to consider the notion of G-algebra in the case of inﬁnite group G. The key assumption in our attempt is to consider subgroups of G which have ﬁnite index. In fact, these types of subgroups play a signiﬁcant part in the theory of groups. Also, the action of the group G on certain co-sets leads to interesting results such as the following old results.

Lemma 1.12 Let G be an inﬁnite group with a subgroup H of ﬁnite index, say n. Then considering the action of G on the co-sets of H, we can identify the factor group G/K with a subgroup of the symmetric group Sn. Here K is the kernel of the permutation representation which is associated with the action. Group action on quantum ﬁeld theory 173

Lemma 1.12 says that for any inﬁnite group G with a subgroup H of ﬁnite index, there is a normal subgroup K of G such that K ≤ H and the factor group G/K is ﬁnite. Therefore, the action of the ﬁnite group G/K on certain objects can be used to deﬁne an action of G in such objects.

Lemma 1.13 Any inﬁnite simple group cannot have a proper subgroup of ﬁ- nite index.

By Lemma 1.13, inﬁnite simple groups will not appear in our work that is because the transfer map cannot be deﬁned. It is well known that groups in QFT are not simple, such as general linear groups, special linear groups and Symplectic groups.

The following results guarantees that there are only ﬁnitely many subgroups of ﬁnite index in certain inﬁnite groups.

Lemma 1.14 For any inﬁnite group which is ﬁnitely generated, for each pos- itive integer n, G has only ﬁnitely many subgroups of index n.

Lemma 1.15 Let H and K be two subgroups of ﬁnite index of a group G, then the intersection subgroup of H and K has also ﬁnite index.

Proof: Consider the index [G : H] = n < ∞ and [G : K] = m < ∞. We will use the Orbit Stabilizer Theorem as in [7]. Deﬁne X as the left co- sets of K, then X = g1K, g2K, ..., gmK. If g1 = 1, then 1 · K = K =⇒ X = K =⇒ the orbit O(K) ⊂ X which means the number of elements of the orbit in K is ﬁnite. Now H acts on X by left multiplication, and StabH (K) = {h ∈ H : hK = K} ⊂ K. Then, [H : H ∩ K] = m. But we know that [G : H] = n then,

nm = [G : H][H : H ∩ K] , which yields to nm = [G : H ∩ K] .

Lemma 1.15, is called Poincar´etheorem for a subgroup of ﬁnite index which can be stated also in the form: Let G be a group and H < G such that [G : H] < ∞. There exists a subgroup N/G such that [G : N] < ∞.

Lemma 1.16 Let H be a subgroup of the group G. If H has ﬁnite index, then the conjugate subgroup Hg also has the same ﬁnite index for each element g ∈ G. 174 Ahmad M. Alghamdi and Roa M. Makki

Proof: Let [G : H] = n < ∞ and [G : Hg] = m < ∞. In order to show that n = m, we deﬁne the left co-sets of H and Hg for a ﬁxed g ∈ G respectively as follows: L = {xH : x ∈ G} ,Lg = {xHg : x ∈ G}. Deﬁne a map φ : L → Lg by φ(xH) = xgHg where xg = g−1xg. This map is well-deﬁned since for xH = yH =⇒ y−1xH = H =⇒ y−1x ∈ H. But g−1(y−1x)g ∈ Hg =⇒ g−1y−1g g−1xg ∈ Hg =⇒ xgHg = ygHg. It is also injective, because if we assume φ(xH) = φ(yH) =⇒ xgHg = ygHg =⇒ (yg)−1xg ∈ Hg. But (yg)−1xg = g−1(yg)−1g g−1xgg = g−1(yg)−1xgg = (y−1x)g =⇒ y−1x ∈ H. Hence, xH = yH. Therefore, the map φ is a surjective map. This means that the number of co-sets in L equals the number of co-sets in Lg and the lemma follows.

2 Brauer Quotient type space

In this section, we shall consider the image space of the transfer map and then form the associated quotient space. Aiming to parallel some well known results in Brauer theory. Let G be a inﬁnite group. For each subgroup H of ﬁnite G index in G, we see that TH (V ) ⊂ InvG(V ). It follows that X G TH (V ) H is a subspace of InvG(V ). In other words,

X G TH (V ) 6 InvG(V ) H and the sum is over all subgroups with ﬁnite index. We shall deﬁne a new quotient space, say V (G) such that:

InvG(V )/I(G) := V (G), where X G I(G) = TH (V ). H

As an extreme case, V (G) = 0 whenever, InvG(V ) = I(G). Lemma 2.1 V (G) is a G-space by an action which is induced via the action of G on V . Proof: Since I(G) is invariant under the action coming from G, we can deﬁne the action on V (G) as follows: for all g in G, for all a + I(G) in V (G), we have (a + I(G))· g = ag + I(G). The conditions for the action is straightforward. Group action on quantum ﬁeld theory 175

Lemma 2.2 V (G) has basis consisting of linear combinations of the orbits of the action of G on V . Proof: The orbits are linearly independent set as they form a partition and clearly they span all the space. We see that the basis of V (G) is permuted by the group G. We call V (G) the permutation Brauer G-space. The next subsection will deal with this space, but with more structure. That is we shall consider a multiplication in our space and then we shall deal with an algebra over a ﬁeld. The results are similar, but here we see some types of ideals in the ﬁxed point subalgebras.

2.1 Group Action on Algebra In this section, we shall continue to assume that our group G to be an inﬁnite group and our class of subgroups of G that we are interesting in is the class of all subgroups of G with ﬁnite index. However, that G will be assumed to act on an algebra over a commutative ring rather than acting on a vector space as in Section 1. The main visible reason that we have a multiplication in our algebra and such action respects that multiplication. Let us recast the deﬁnition of an algebra over a commutative ring as well as the action of a group G on that algebra.

Deﬁnition 2.3 A ring A in which it has a suitable module structure over a commutative ring R with identity is called an algebra over R if it satisﬁes

r(ab) = (ra)b = a(rb), for all r ∈ R and a, b ∈ A.

In fact, the above deﬁnition of algebra is equivalent to the existence of a suitable ring homomorphism from the ring R to the center of A. Now we state the deﬁnition of G-algebra over a commutative ring R where G is any group which is not necessarily to be ﬁnite. Deﬁnition 2.4 Let G be a group. An algebra A over a commutative ring R with identity is called an algebra over R if there is a group homomorphism from the group G to the group of automorphisms of A. Deﬁnition 2.4 is equivalent to the existence of an action from the group G to on the algebra A such that A is a G-space as in Deﬁnition 1.1 and for each g ∈ G and for each a, b ∈ A,

(ab)g = agbg.

The new construction enables us to deal and tackle with G-ideals, G- quotient as well as G-algebra homomorphisms. As examples of G-algebras, 176 Ahmad M. Alghamdi and Roa M. Makki we mention the group algebra as well as the algebra of endomorphisms. See more details in the books [5, 9, 2]. The ﬁrst interesting object of a G-algebra setting over a ring is the G-ﬁxed points. That is the set of all elements in the G-algebra A which are ﬁxed under G action. Namely, a ∈ A such that ag = a for all g in G. As a notation, we refer to that set by AG. Note that AG may or may not coincide with the center of A. The algebraic structure of AG is very important. In fact, it is a G-subalgebra which contains many important G-ideals such as Reynolds ideal. Also, we mention that for each H subgroup in G, we have AG ⊂ AH . In particular, we see that AG ⊂ AH ⊂ AK ⊂ A, for any chain 1G ≤ K ≤ H ≤ G. We emphasise that our group under consideration can be an inﬁnite group. The treatment of these G-algebras goes back to Green [4], but for ﬁnite groups and their subgroups. Our main consideration in this work as we mentioned already in the introduction is to get some sort of generalization of these theory in the case of inﬁnite groups. We choose quantum ﬁeld theory to try to do some applications. The main restriction we shall do is that subgroups of G need to be subgroups of ﬁnite index. The reason for such choice is to make the G following sum of the relative trace map TH has a meaning in the G-algebra A. We mean that a similar application can be used for G-algebras for G an inﬁnite group. Namely, the relative trace map

G H G TH : A → A such that for each a ∈ AH ,

G X g TH (a) = a , g∈T

G where T is the transversal set. It follows that map TH is linear and independent of our choice of the transversal set that we sum over. But it is not multiplicative in general. Here, that H is a subgroup of G with ﬁnite index. The condition of the index gives us the opportunity to consider that sum over the transversal set which contains ﬁnite numbers of element. There are many properties for the relative trace map which are somehow similar to that in Section 1. But the most important one the so called Frobenius lemma as follows: Group action on quantum ﬁeld theory 177

Lemma 2.5 The image of the relative trace map is an ideal in the ﬁxed point under the action by the whole group. The ideal in Lemma 2.5 is very important in many investigations. In fact, it has speciﬁc name namely Reynolds ideal.

3 Group Action on Quantum Field Theory

This section is devoted to space time setting which is the main topic in quantum ﬁeld theory. We may now apply our concept of group action on quantum ﬁeld theory by considering G to be the Poincar´egroup and V to be the Minkowski space. Deﬁnition 3.1 The 3-dimensional real Euclidian space with spacial coordi- nates (x1 = x, x2 = y, x3 = z) and a one time dimensional real vector space with coordinate (x0 = ct) where c is the speed of light, can be joined to give the so-called Minkowski space time which we denote by M. It is common to set the speed of light c to be equals 1. Therefore, we are dealing with a 4-dimensional real vector space equipped with space time interval written as ds2 = dt2 − dx2 − dy2 − dz2, and a Minkowski metric which is given in matrix form by 1 0 0 0 0 −1 0 0 g = . 0 0 −1 0 0 0 0 −1

Note that g−1 = g. For more details, the reader can see the reference [8]. Our main concern is to apply the transfer map and abstract group action in this ﬁeld. Now we shall repeat the construction in Section 1 for this space. In fact, there are many groups which act on Minkowski space such as its automorphism, the Lorentz group.

Lemma 3.2 Let G be a group that acts on Minkowski space time M and H is a subgroup of G, then for all g ∈ G, we have

InvHg (M) = InvH (M).g

Proof: Follows directly from Section 1.

Lemma 3.3 Let G be a group that acts on Minkowski space time M. If G G H,K are subgroups of G, then TH (M) ⊆ TK (M). In particular, if K is a G G G-conjugate subgroup H, then TH (M) = TK (M). 178 Ahmad M. Alghamdi and Roa M. Makki

Proof: This lemma is a special case of the construction in Section 1. Now we shall deﬁne a new construction in QFT which we call the Minkowski quotient space time and as notation can be written as M(G). Deﬁnition 3.4 The Minkowski quotient space time is

InvG(V )/I(G) := M(G), where, X G I(G) = TH (M). H

For a group G and two subgroups H and K of that group G, one also can deﬁne a relation on G such that for all x, y ∈ G; x is related to y if and only if y is an element in HxK. It follows that such relation is an equivalence relation. This equivalence relation generates equivalence classes which is called double co-sets. Lemma 3.8 Double co-sets are ﬁnite for any two subgroups of ﬁnite index. The following theorem is the main contribution in this work regarding the relationship with QFT. We call it Mackey Decomposition Theorem for QFT. Theorem 3.9 Let G be a group containing two subgroups H and K of ﬁnite index. Let M be a Minkowski space which receives an action from the subgroup K. That is M is a K-space. Then we have the following isomorphism

G G ∼ M H Kg ResH (IndK (M)) = IndH∩Kg (ResH∩Kg (M ⊗ g)). HgK The sum here is over the double co-sets HgK. Group action on quantum ﬁeld theory 179

Proof: By Lemma 3.8, the sum in the right hand side is well deﬁned and as spaces, both sides coincides. It remains to check the action. However, the induction followed by restriction is well known and fundamental theorem in representation theory [2] which is called Mackey Decomposition Theorem.

References

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Received: July 31, 2019, Published: August 20, 2019