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Cambridge Monographs on Mathematical Physics
Introduction to Supersymmetry
Just as ordinary symmetries relate various forms of matter to each other, and various basic forces to each other, so the novel concept of supersymmetry relates (Fermi) matter to (Bose) force. It is the aim of this book to provide a brief introductory description of the new physical and mathematical ideas involved in formulating supersymmetric theories. The book starts with a physical motivation of supersymmetry, a presentation of the mathematics of Lie superalgebras, supergroups and superspace. Techniques for constructing manifestly globally supersymmetric field theories are given, using the superfield formalism. To allow for a clear flow of ideas, the basic ideas and techniques are worked out in low space dimensionalities where the formulae do not obscure the concepts. Generalizations to four space-time dimensions are then readily come by. Some quantum aspects are discussed. Possible phenomenological applications are not emphasized. Supergravities, locally supersymmetric theories are then considered in 4 and 11 dimensions, in component formalism. An introduction to supersymmetry will be of interest to postgraduate students and researchers in theoretical and particle physics, especially those working in quantum field theory, quantum gravity, general relativity and supergravity. The book will also be of interest to mathematicians with an interest in theoretical physics.
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First published January 1, 1986
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April 5, 2024
AN ‘INTRODUCTORY TEXT’ (ALBEIT VERY TECHNICAL) ABOUT SUPERSYMMETRY
Peter George Oliver Freund (1936- 2018) was a professor of theoretical physics at the University of Chicago.
He wrote in the Preface to this 1986 book, “Supersymmetry is one of the boldest, most original and most fruitful ideas to appear in physics in a very long time… nobody doubts that when discarding some blinding prejudices, or coming by some new data, supersymmetry will come into its own, experimentally as well… it has created its own, rich, truly new mathematics. Yes, we are faced here with one of those rare instances, when the mathematicians, in all their wisdom, have overlooked a beautiful and most useful structure, and come to appreciate it only at the demand of physicists. We are living in an era in which the contacts between mathematics and physicists are being vigorously renewed… This is a good omen, since such contacts have historically always led to great advances both in mathematics and physics… While teaching a course on supersymmetry … it became clear to me that there was still ample space left for a brief introductory text… I included my own assessment on the present status of supersymmetry, of where the field seems to be heading.”
He explains, “Quantum field theories, even of the tame renormalizable type, contain (manageable) divergences in perturbation theory. As a rule, supersymmetric theories also contain such divergences, but less virulent ones than in nonsupersymmetrical theories of similar type. Indeed, after years of getting used to the idea that divergences in quantum field theory are simply unavoidable, supersymmetric theories, possibly even some realistic ones, have been produced in which all divergences cancel to all orders in perturbation theory.” (Pg. 81) Later, he adds, “the customary divergences of quantum field theories getting alleviated by supersymmetry. Over the last few years it has been established that in the case of extended supersymmetry there exist theories for which the divergences are not only alleviated but outright eliminated: the theories are finite, no divergencies whatsoever!” (Pg. 84)
He acknowledges, “The fundamental symmetries exhibited by the laws of nature are local rather than global, i.e. they are gauged… If the laws of nature turn out to exhibit some form of supersymmetry, it is then natural to ask whether supersymmetry can also be gauged.” (Pg. 97) He continues, “constructing supergravity in a supermanifold formalism is not a straightforward geometric task. To be sure, such a construction is possible, but in its present form sufficiently complicated to foster the belief that the final word on superspace-supergravity has not yet been spoken.” (Pg. 98)
He states, “There are some features of N=8 supergravity, shared… with higher-dimensional supergravities, that make these theories so very attractive. First of all, there are some assumptions in both ordinary and supersymmetric grand unification, so well hidden, that it is usually glossed over that they even are assumptions… No previous physical theory has exhibited anything like this degree of self-containedness and completeness. It is in view of all this that attempts at constructing a phenomenologically viable… supergravity, or superstring theory are being intensely pursued at present. This brings us unambiguously to higher-dimensional theories.” (Pg. 117-119)
He asks, “Somehow this all flies in the face of an unwritten rule of theoretical physics, namely that important theories be simple, unique and beautiful. Could it be that the apparent aesthetic flaws of these theories, are consequences of our way of looking at them, rather than of the theories themselves? What I have I mind is something like looking at an animal outside of its natural habitat when it can easily appear clumsy and weird. Only replacing it on its home ground will reveal its natural grace.” (Pg. 120)
He observes, “We are then confronted with the questions: (I1) what is the ‘true’ dimension of space-time? (II) Is there a way of predicting the ‘true’ and the ‘apparent’ dimensions of space-time?... the most compelling reason to consider higher-dimensional theories comes from the need to construct a consistent quantum theory of gravity. Already in four space-time dimensions, gravity is nonrenormalizable. The only way to achieve quantum consistency for gravity then remains the possibility of a finite quantum theory: infinitely many counterterms may be needed but their coefficients are calculable and finite (small) rather than arbitrary constants.” (Pg. 121-122)
He states, “If time is among the four dimensions then the dynamics is right to compactify the remaining seven dimensions. It is now clear that their preferential 4+7 split of the originally eleven-dimensional space-time can be traced directly to supersymmetry. It is thus supersymmetry that ‘dials’ the dimensionality of the observed space-time. Yet as it stands, this argument still has some weak spots and outright difficulties. First of all the time dimension could be among the seven rather than the four.” (Pg. 130)
He suggests, “If this finiteness is to hold up at higher loops, as seems now virtually certain, then a combination of supersymmetry … and of the Veneziano-Nambu string idea may finally have achieved the long sought-after synthesis of quantum theory and general relativity. At the same time supersymmetry will have returned to string theory where it was originally conceived. All this would be yet another example of a phenomenon often encountered in physics: a theory, originally pursued for its mathematical and philosophical beauty, is shown to possess a new feature which then turns into the driving force of research into the theory. Renormalizability …was this new feature for electroweak theory, finiteness in the presence of gravity (?) may just be it for supergravity and superstring type theories.” (Pg. 133-134)
He summarizes, “It may be appropriate to conclude this book by assessing the present status of the principle of supersymmetry in physics. A physical principle can be reliably evaluated according to the following criteria: (A) The experimental evidence that supports the principle. (B) New phenomena predicted on the basis of the principle. (C) The experimental and theoretical puzzles solved by the principle. (D) The internal consistency of theories that incorporate the principle. (E) The aesthetic and philosophic advantages of the principle. As the first three parts of this book imply, supersymmetry fares poorly on criterion (A). There is no hard evidence for supersymmetry in particle physics (at the time of this writing). Given the mathematical novelty of the concept, one may wonder whether supersymmetric systems appear in physics at all. There are two fields of physics in which supersymmetry does indeed make an appearance: the quantum statistical mechanics of two-dimensional systems and nuclear physics.” (Pg. 137)
He continues, “On criterion (C), supersymmetry has not solved any experimental puzzles, partly because for a long time there just hadn’t been any such puzzles around! Concerning theoretical puzzles, the hierarchy problem is certainly an important and venerable puzzle… Here… supersymmetry is of use. Further problems such as why space-time appears four-dimensional, and the force/matter problem also receive a first meaningful treatment in the framework of supersymmetry. Based on criterion (C) then, supersymmetry does make an impact. On criterion (D) supersymmetry fares very well indeed… On criterion (E) supersymmetry is also a clear winner. New mathematical structures have emerged: superalgebras, supergroups, supermahifolds. A much more unified picture is possible…and again we can refer to the resolution of the age old force/matter problem. It is after all to a large extent these aesthetic, mathematical and philosophical merits that have driven much of the early work on supersymmetry.” (Pg. 139)
He concludes, “It may well be that we already possess all the essential ingredients for a successful implementation of supersymmetry in particle physics, but are blinded by an assumption as obvious to ‘us all,’ as Sakurai’s ideas about flavor gauging were in their time. One of the hardest steps in science is the discarding of ‘obvious’ flaws. Let me not speculate in writing about what the possible obvious flaws in present-day supersymmetry ideas may be. After all, there is always the obvious way out: not enough accelerator energy. Time will tell.” (Pg. 141)
This is a VERY ‘technical’ book, that will mostly be of interest to those seriously studying the mathematics, etc., of supersymmetry.
Peter George Oliver Freund (1936- 2018) was a professor of theoretical physics at the University of Chicago.
He wrote in the Preface to this 1986 book, “Supersymmetry is one of the boldest, most original and most fruitful ideas to appear in physics in a very long time… nobody doubts that when discarding some blinding prejudices, or coming by some new data, supersymmetry will come into its own, experimentally as well… it has created its own, rich, truly new mathematics. Yes, we are faced here with one of those rare instances, when the mathematicians, in all their wisdom, have overlooked a beautiful and most useful structure, and come to appreciate it only at the demand of physicists. We are living in an era in which the contacts between mathematics and physicists are being vigorously renewed… This is a good omen, since such contacts have historically always led to great advances both in mathematics and physics… While teaching a course on supersymmetry … it became clear to me that there was still ample space left for a brief introductory text… I included my own assessment on the present status of supersymmetry, of where the field seems to be heading.”
He explains, “Quantum field theories, even of the tame renormalizable type, contain (manageable) divergences in perturbation theory. As a rule, supersymmetric theories also contain such divergences, but less virulent ones than in nonsupersymmetrical theories of similar type. Indeed, after years of getting used to the idea that divergences in quantum field theory are simply unavoidable, supersymmetric theories, possibly even some realistic ones, have been produced in which all divergences cancel to all orders in perturbation theory.” (Pg. 81) Later, he adds, “the customary divergences of quantum field theories getting alleviated by supersymmetry. Over the last few years it has been established that in the case of extended supersymmetry there exist theories for which the divergences are not only alleviated but outright eliminated: the theories are finite, no divergencies whatsoever!” (Pg. 84)
He acknowledges, “The fundamental symmetries exhibited by the laws of nature are local rather than global, i.e. they are gauged… If the laws of nature turn out to exhibit some form of supersymmetry, it is then natural to ask whether supersymmetry can also be gauged.” (Pg. 97) He continues, “constructing supergravity in a supermanifold formalism is not a straightforward geometric task. To be sure, such a construction is possible, but in its present form sufficiently complicated to foster the belief that the final word on superspace-supergravity has not yet been spoken.” (Pg. 98)
He states, “There are some features of N=8 supergravity, shared… with higher-dimensional supergravities, that make these theories so very attractive. First of all, there are some assumptions in both ordinary and supersymmetric grand unification, so well hidden, that it is usually glossed over that they even are assumptions… No previous physical theory has exhibited anything like this degree of self-containedness and completeness. It is in view of all this that attempts at constructing a phenomenologically viable… supergravity, or superstring theory are being intensely pursued at present. This brings us unambiguously to higher-dimensional theories.” (Pg. 117-119)
He asks, “Somehow this all flies in the face of an unwritten rule of theoretical physics, namely that important theories be simple, unique and beautiful. Could it be that the apparent aesthetic flaws of these theories, are consequences of our way of looking at them, rather than of the theories themselves? What I have I mind is something like looking at an animal outside of its natural habitat when it can easily appear clumsy and weird. Only replacing it on its home ground will reveal its natural grace.” (Pg. 120)
He observes, “We are then confronted with the questions: (I1) what is the ‘true’ dimension of space-time? (II) Is there a way of predicting the ‘true’ and the ‘apparent’ dimensions of space-time?... the most compelling reason to consider higher-dimensional theories comes from the need to construct a consistent quantum theory of gravity. Already in four space-time dimensions, gravity is nonrenormalizable. The only way to achieve quantum consistency for gravity then remains the possibility of a finite quantum theory: infinitely many counterterms may be needed but their coefficients are calculable and finite (small) rather than arbitrary constants.” (Pg. 121-122)
He states, “If time is among the four dimensions then the dynamics is right to compactify the remaining seven dimensions. It is now clear that their preferential 4+7 split of the originally eleven-dimensional space-time can be traced directly to supersymmetry. It is thus supersymmetry that ‘dials’ the dimensionality of the observed space-time. Yet as it stands, this argument still has some weak spots and outright difficulties. First of all the time dimension could be among the seven rather than the four.” (Pg. 130)
He suggests, “If this finiteness is to hold up at higher loops, as seems now virtually certain, then a combination of supersymmetry … and of the Veneziano-Nambu string idea may finally have achieved the long sought-after synthesis of quantum theory and general relativity. At the same time supersymmetry will have returned to string theory where it was originally conceived. All this would be yet another example of a phenomenon often encountered in physics: a theory, originally pursued for its mathematical and philosophical beauty, is shown to possess a new feature which then turns into the driving force of research into the theory. Renormalizability …was this new feature for electroweak theory, finiteness in the presence of gravity (?) may just be it for supergravity and superstring type theories.” (Pg. 133-134)
He summarizes, “It may be appropriate to conclude this book by assessing the present status of the principle of supersymmetry in physics. A physical principle can be reliably evaluated according to the following criteria: (A) The experimental evidence that supports the principle. (B) New phenomena predicted on the basis of the principle. (C) The experimental and theoretical puzzles solved by the principle. (D) The internal consistency of theories that incorporate the principle. (E) The aesthetic and philosophic advantages of the principle. As the first three parts of this book imply, supersymmetry fares poorly on criterion (A). There is no hard evidence for supersymmetry in particle physics (at the time of this writing). Given the mathematical novelty of the concept, one may wonder whether supersymmetric systems appear in physics at all. There are two fields of physics in which supersymmetry does indeed make an appearance: the quantum statistical mechanics of two-dimensional systems and nuclear physics.” (Pg. 137)
He continues, “On criterion (C), supersymmetry has not solved any experimental puzzles, partly because for a long time there just hadn’t been any such puzzles around! Concerning theoretical puzzles, the hierarchy problem is certainly an important and venerable puzzle… Here… supersymmetry is of use. Further problems such as why space-time appears four-dimensional, and the force/matter problem also receive a first meaningful treatment in the framework of supersymmetry. Based on criterion (C) then, supersymmetry does make an impact. On criterion (D) supersymmetry fares very well indeed… On criterion (E) supersymmetry is also a clear winner. New mathematical structures have emerged: superalgebras, supergroups, supermahifolds. A much more unified picture is possible…and again we can refer to the resolution of the age old force/matter problem. It is after all to a large extent these aesthetic, mathematical and philosophical merits that have driven much of the early work on supersymmetry.” (Pg. 139)
He concludes, “It may well be that we already possess all the essential ingredients for a successful implementation of supersymmetry in particle physics, but are blinded by an assumption as obvious to ‘us all,’ as Sakurai’s ideas about flavor gauging were in their time. One of the hardest steps in science is the discarding of ‘obvious’ flaws. Let me not speculate in writing about what the possible obvious flaws in present-day supersymmetry ideas may be. After all, there is always the obvious way out: not enough accelerator energy. Time will tell.” (Pg. 141)
This is a VERY ‘technical’ book, that will mostly be of interest to those seriously studying the mathematics, etc., of supersymmetry.
Displaying 1 of 1 review