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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 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 action on a and on algebra by using 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 and then we have the notion of a group acts on a set see [7]. Such object can be a which results the so called G-space or G- 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 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 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 of that object or a subgroup of it. Let us remind the reader that the 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 . 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 . 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 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 , 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 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 Sn. Here K is the of the 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 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 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 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 . 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 .

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

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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[2] C. Curtis, I. Reiner, Representation Theory of Finite Groups and As- sociative Algebras, Pure and Applied Mathematics, Vol. XI Interscience Publishers, a division of John Wiley and Sons, New York-London 1962.

[3] D. Gorenstein, Finite Groups, AMS Chelsea Publishing, 2nd Edition, 1982. https://doi.org/10.1007/978-1-4684-8497-7

[4] J. A. Green, Some remarks on defect groups, Math. Z., 107 (1968), 133- 150. https://doi.org/10.1007/bf01111026

[5] B. K¨ulshammer, Lectures on Block Theory, Cambridge University Press, Cambridge, 1991. https://doi.org/10.1017/cbo9780511565786

[6] H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, 1989.

[7] J. Rose, A First Course on , Cambridge University Press, 1978 (reprinted by Dover 1994).

[8] L.H. Ryder, Quantum Field Theory, Cambridge University Press, 1985. https://doi.org/10.1017/cbo9780511813900

[9] J. Th´evenaz, G-Algebras and Modular Representation Theory, Oxford University Press, Oxford, 1995.

[10] A. Tucker, Applied Combinatorics, 4th Edition, John Wiley and Sons, 2002.

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