대칭성과 대칭성 붕괴 (Symmetry and Symmetry Breaking)

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참조 문헌

[1] Symmetry and Symmetry Breaking, The Stanford Encyclopedia of Philosophy (Fall 2023 Edition), Brading, Katherine, Elena Castellani, and Nicholas Teh, Edward N. Zalta & Uri Nodelman (eds.)

0. INTRODUCTION

　대칭성에 대한 논의는 양자이론과 상대성이론, 두 현대 물리학을 지배하고 있다. 학자들은 gauge symmetry의 중요성, permutation symmetry를 고려했을 때의 quantum particle identity, parity violation을 어떻게 이해해야 하는지, symmetry breaking의 역할, the empirical status of symmetry principles 등과 같은 문제들에 점점 더 많은 관심을 기울이기 시작하고 있다. 이러한 이슈들은 과학의 철학에서 언급되는 전통적인 문제들과 직접적으로 연결된다. 이 점에는 status of the law of nature, 그리고 수학, 물리 이론, 자연 사이의 관계들이 포함된다.

해당 문헌은 현대 물리학에서 이용되고 있는 대칭성에 대한 개념의 역사적 근원과 발생을 간단하게 설명하면서 시작한다. 그리고 이 물리적 개념을 이용한 응용으로 눈을 돌려 대칭성이 사용되는 두가지 서로 다른 용례에 대해 구분하고자 한다: Symmetry Principles vs Symmetry Arguments. 이는 물리 분야에서 사용되는 다양한 물리적 대칭성의 종류를 소개하며, 이들이 어떻게 물리학에 도입되었는지를 설명한다. 그리고는 이 다양한 대칭성에 대한 디테일한 부분에서 한발자국 물러서서 물리학에서의 대칭성의 중요성과 상태를 기술하는 전반적인 nautre에 대해 중점을 둘것이다.

1. THE CONCEPT OF SYMMETRY

　"Symmetry"라는 용어는 고대 그리스 언어 sun (with, 또는 togeher를 뜻 함)과 metron (measure를 뜻함)의 합성어 symmetria에서 유래 됐으며, 원래는 commensurability(공약성, 두 양(quantity)을 동일한 척도나 단위를 사용해 측정할 수 있음)의 뜻을 가졌다.

이 용어는 이후 더 일반적인 의미를 얻게 됐는데, It quickly acquired a further, more general, meaning: that of a proportion relation, grounded on (integer) numbers, and with the function of harmonizing the different elements into a unitary whole. From the outset, then, symmetry was closely related to harmony, beauty, and unity, and this was to prove decisive for its role in theories of nature. In Plato’s Timaeus, for example, the regular polyhedra are afforded a central place in the doctrine of natural elements for the proportions they contain and the beauty of their forms: fire has the form of the regular tetrahedron, earth the form of the cube, air the form of the regular octahedron, water the form of the regular icosahedron, while the regular dodecahedron is used for the form of the entire universe. The history of science provides another paradigmatic example of the use of these figures as basic ingredients in physical description: Kepler’s 1596 Mysterium Cosmographicum presents a planetary architecture grounded on the five regular solids.

From a modern perspective, the regular figures used in Plato’s and Kepler’s physics for the mathematical proportions and harmonies they contain (and the related properties and beauty of their form) are symmetric in another sense that does not have to do with proportions.

현대 과학의 언어로 볼 때, 정다각형이나 정다면체와 같은 기하학적 figure의 대칭성은 특정 rotation이나 reflection 그룹에 대한 invariance를 기준으로 정의된다. 이러한 정의는 어디에서 왔는가? 고대 그리스와 로마에서 사용됐던 대칭성이라는 정의에 더해, 이와는 또 다른 정의의 대칭성이 17세기 무렵에 나타나기 시작했다. 이때의 대칭성은 고대의 대칭성이 얘기하는 proportion과는 다르게, 서로 대립되는 요소들 (figure의 왼쪽 부분과 오른쪽 부분) 간의 equality relation에 기초한다.

이 중 특히 중요한 점은, 각 요소들이 전체의 관점에서 서로 교환(exchange) 가능하다는 것이다. 이들은 서로 교환해도 원래의 figure 자체는 그대로 유지된다. 이러한 대칭성 개념은 여러 단계를 거쳐 발전하면서, 오늘날 현대 과학에서 널리 사용되는 핵심 개념으로 자리 잡았다. 특히 중요한 발전 단계 중 하나는 특정 수학적 연산의 도입이었는데, 여기에 반사(reflection), 회전(rotation), 평행이동(translation) 등이 포함되며, 이를 통해 어 떠한 요소들이 교환 가능한지 정확히 묘사한다. 결과적으로, 우리는 기하학적 figure의 대칭성을 “지정된 연산들 중 하나에 따라 동일한 구성 요소들을 교환했을 때에도 figure는 변하지 않는다”는 불변성(invariance)을 기준으로 정의하게 된다.

Thus, when the two halves of a bilaterally symmetric figure are exchanged by reflection, we recover the original figure, and that figure is said to be invariant under left-right reflections. This is known as the “crystallographic notion of symmetry”, since it was in the context of early developments in crystallography that symmetry was first so defined and applied.[1]

다음 단계는 이 개념을 generalization 하여 group-theoretic definition of symmetry로 확장하는 것이었다. 이 개념은 19세기에 발전한 군(group)의 대수적 개념에 기반하여 부각되기 시작했으며, 도형의 대칭 연산들이 군을 형성하기 위한 조건을 만족한다는 사실이 확인되었다. 예를 들어, reflection symmetry는 이제 invariance under the group of reflection이라는 용어로 명확히 정의된다. 결과적으로, 우리는 symmetry, equivalence, 그리고 group 사이에 매우 긴밀한 연결관계를 갖게 된다: symmetry group은 equivalence classes에 partition을 유도한다.

For example, reflection symmetry has now a precise definition in terms of invariance under the group of reflections. Finally, we have the resulting close connection between the notion of symmetry, equivalence and group: a symmetry group induces a partition into equivalence classes. The elements that are exchanged with one another by the symmetry transformations of the figure (or whatever the “whole” considered is) are connected by an equivalence relation, thus forming an equivalence class.[3]

The group-theoretic notion of symmetry is the one that has proven so successful in modern science. Note, however, that symmetry remains linked to beauty (regularity) and unity: by means of the symmetry transformations, distinct (but “equal” or, more generally, “equivalent”) elements are related to each other and to the whole, thus forming a regular “unity”. The way in which the regularity of the whole emerges is dictated by the nature of the specified transformation group. Summing up, a unity of different and equal elements is always associated with symmetry, in its ancient or modern sense; the way in which this unity is realized, on the one hand, and how the equal and different elements are chosen, on the other hand, determines the resulting symmetry and in what exactly it consists.

“Invariance under a specified group of transformations” 으로서의 대칭성의 정의는 이 개념이 더 다양하게, 단순히 공간적인 도형에 대해서 뿐만 아니라 수학적 표현과 같은 추상적인 오브젝트에 대해서도, 응용될수 있게 만들었다. 특히, 동역학과 같은 물리적인 표현에 대해서도 대칭성을 사용할 수 있게 됐다. 더욱이, group theory를 물리학 이론에 도입함으로써, 이를 활용할 수 있게 되었다.

역사적 관점에서 물리학에서의 대칭성의 역할을 고려 했을때, 두개의 distinction을 숙지하는 것이 좋다.

첫번째는 이 개념에 대한 implicit, 그리고 explicit 사용례이다. 대칭성에 대한 논의는 그동안 자연을 기술하는데 적용 됐지만, 오랫동안 주로 implicity 한 방법으로 기술 됐다. 우리가 봐왔듯이, (우리가 여기서 관심있어 하는) 과학적 관점으로서의 대칭성은 오래 되지 않은 것이다. 만약 우리가 자연을 기술하는 고대의 이론 에서 대칭성의 개념을 얘기한다면, 우리는 반드시 이것이 implicit 한 방법으로 기술 됐음을 명확히 해야 한다.
두번째는 대칭성을 사용하는 두가지 주요 방법이다.
(1) 우리는 phenomena나 laws에 특정한 대칭성 특성을 부여할 수 있다 이것은 오브젝트나 현상이라기 보다는 법칙과
관련된 application으로, 현대 물리학의 중점이 됐다. = SYMMETRY PRINCIPLES
(2) 우리는 특정 물리적 상황이나 현상과 관련된 특정 결과를, 그들의 대칭성 특성을 기반하여 끄집어 낼 수 있다.
= SYMMETRY ARGUMENTS

2. SYMMETRY PRINCIPLES

　 물리학 방정식의 불변적 특성에 대한 연구가 처음으로 수면위로 올라온건 1800년대 초-중반이었으며 고전역학의 framework에서 움직임에 관한 문제를 변형시키려는 것이 시초였다. W. R. Hamilton이 창안한 동역학적 방정식의 공식화(Hamiltonian, 또는 canonical formulation 이라 한다)를 이용하여, C. H. Jacobi는 Hamiltonian 방정식은 불변하되, 변수를 transformation시키는 전략으로 운동 방정식의 해에 도달하는 방법을 만들어냈다. 각 단계 마다 transformation이 수행되면서 기존의 문제는 더욱 간단해지지만 완벽하게 동등한 방정식을 유지한채 말이다. 이 Jacobi의 canonical tranformation theory(비록 동역학 문제를 '단순히 기계적으로 푸는 방법'으로 소개되었지만)는 매우 중요한 발자취를 남겼다: 바로 물리학적 이론을 transformation property 관점에서 하는 연구의 시작이다. 이에 대한 예로서는 canonical transformation에서의 불변성에 대한 연구가 있으며 Poisson brackets, Poincaré’s integral invariants도 역시 그 예시중 하나이다. 그리고 시간이 흘러, 물리적 불변량에 대한 연구와 불변량에 대한 대수적, 기하학적 이론 사이의 접점은 1900년대 중후반기에 번영하기 시작했으며, 동역학적 문제에 대한 기하학적 접근의 토대를 태동시킨다.

20세기 초반 Gottingen에서, F. Klein, D. Hilbert, H. Weyl, 그리고 나중에 가서는 E.Noether까지 쟁쟁한 물리학자 그룹이 물리 이론을 위해 group theory 수학을 사용하였다. 이때의 발전사에 대한 자세한 내용은 Brading and Castellani (2007)를 참조하기 바란다.

위의 접근법에서는, 우리가 관심있어 하는 물리적 방정식 또는 표현이 주어진 상태이며, 이때의 symmetry 성질을 학습하는 것이 전략이었다. 하지만, 또 다른 접근법이 존재하는데, 이름하여 'reverse one'이다. 이는 특정 대칭성에서부터 시작하여 이 대칭성을 만족하는 동역학적 방정식을 찾는 전략이다. 다시 말해, 동역학적 방정식에서부터 그 대칭성을 끄집어내는게 아니라, 애초에 해당 대칭성이 물리적으로 중요하다고 미리 상정해놓는 것이다. 물론, 특정 대칭성이 자연에 존재한다는 가정은 novelty 하지 않다. 비록 대칭성의 법칙에 분명하게 표현 되어있지는 않지만, physical space의 homogenity와 isotropy, 그리고 시간의 uniformity는 현대 과학이 태동한 이래로 물리학적 묘사(physical description)의 전제조건으로 여겨져 왔다. 이 종류의 symmetry principle을 의도적으로 사용한 아마도 가장 유명한 예는 1632년 Galileo에 의해 쓰여진 Dialogue concerning the two chief world systems 일 것이다. 여기에서 갈릴레오는 지구가 움직이는가에 대한 의논을 한다. 갈릴레오는 지구에서 물체들이 어떻게 행동하는지(돌이 떨어진다거나, 새가 날아다닌다거나) 그저 보는 것만으로 지구가 돌지 않고 멈춰있다고 주장할 수 없다고 생각했다. 이러한 관측만으로는 지구의 운동상태를 결정할 수 없다는 것이었다.

그의 접근법은 배를 이용한 비유였다: 그는 배의 선실 내부에서 물체의 행동에 대해 생각했다. 만약 선실 밖의 어떤 것도 참조하지 않고 선실 내부에서 수행한 실험만을 수행 한다면, 배가 정지해 있는지, 아니면 지표면을 따라 부드럽게 이동하고 있는지를 구별할 수 없다고 주장했다. 정지 상태와 특정 종류의 운동 상태가 대칭을 이룬다고 가정하면, 선실 내부에서 진행되는 실험을 지배하는 법칙의 세부 사항을 몰라도 이 결과를 예측할 수 있게 되는 것이다. Galilean principle of relativity는 axiom으로써 빠르게 인정 받았고, 17세기에 널리 사용된다. 가장 대표적인 예로 호이겐스는 충돌하는 물체에 대한 문제를 풀때, 그리고 뉴턴은 work of motion에 대한 이른 정의에 이를 사용했다. Huygens took the relativity principle as his 3rd hypothesis or axiom, but in Newton’s Principia it is demoted to a corollary to the laws of motion, its status in Newtonian physics therefore being that of a consequence of the laws, even though it remains, in fact, an independent assumption.

비록 고전 역학 법칙의 공간적, 시간적 불변성이 매우 오랜시간 동안 활용되어 왔고 아인슈타인의 1905년 특수상대성이론 논문 이전에, 이미 전자기학에서의 global spacetime symmetry의 그룹은 푸앙카레에 의해 유도 됐지만, 물리적 법칙에 대한 대칭성의 지위가 뒤바뀐 것은 아인슈타인이 1905년 특수상대성이론 논문을 작성하면서부터였다.

E.P. Wigner (1967, p. 5)는 글에서 다음과 같이 썼다. “그 원리들의 중요성과 일반적 타당성은 오직 아인슈타인에 의해서만 비로소 인정받았다”고 하면서, 특수상대성이론에 대한 아인슈타인의 업적은 “기존 흐름의 전환점을 이룬다. 그 전까지는 불변성 원리가 운동 법칙에서 도출되었으나 … 이제는 반대로 자연의 법칙을 불변성 법칙을 통해 도출하고, 그 타당성을 검증하는 것이 자연스럽게 되었다.”

In postulating the universality of the global continuous spacetime symmetries, Einstein’s construction of his special theory of relativity represents the first turning point in the application of symmetry to twentieth-century physics.[8]

2.1 Relativity

2.1.1 The special theory of relativity

아인슈타인의 특수상대성 이론 (special theory of relativity, STR)은 두개의 근본적 가정을 기반으로 만들어졌다.

The light postulate. : the speed of light, in the “rest frame”, is independent of the speed of the source
The principle of the relativity.

여기서 아인슈타인은 두번째 법칙을 이용하여 명시적으로 법칙의 형태를 제한한다. 따라서 우리는 constructive theory와 principle theory의 차이를 확인할 수 있다. 전자의 경우, 실제 물체들의 구성과 거동에 대한 알려진 사실을 바탕으로 이론을 구축한다면, 후자의 경우는 특정 원리를 채택함으로써 가능한 형태를 미리 제한하는 방식으로 시작한다.

아인슈타인에 의해 적용된 princple of realitvity (1905, p. 395)은 다음과 같이 기술된다. "물리적 시스템의 state가 변화하는 방식을 규정하는 법칙은, 이러한 상태 변화가 서로 균일한 병진 운동을 하는 두 좌표계 중 어느 한 좌표계에 의해 표현되든 상관없이 동일하다. (The laws by which the states of physical systems undergo changes are independent of whether these changes of states are referred to one or the other of two coordinate systems moving relatively to each other in uniform translational motion.)"

이 법칙은, 광속 불변의 법칙(+기타 가정)과 만났을 때, Lorentz transforation을 야기한다.

This principle, when combined with the light postulate (and certain other assumptions), leads to the Lorentz transformations, these being the transformations between coordinate systems moving uniformly with respect to one another according to STR.

STR에 따르면, 물리적 법칙은 Lorentz transfomation에 대해서, 사실 모든 Poincare group of transformation에 대해, invariant하다. 이 transformations들은 고전 역학에서의 Galilean transformations과는 다른 것이다. H. Minkowski는 STR을 재구성하여, 공간과 시간이 서로 합쳐진 4차원 geometry임을 보였다. 이 관점에서는, STR에서 Poincaré group 대칭 변환은 시공간 구조의 일부를 이룬다. 이러한 이유로, 이 대칭성들은 위그너(Wigner, 1967, 특히 15쪽 및 17–19쪽)에 의해 “기하학적 대칭성(geometric symmetries)”으로 명명되었다.

There is a debate in the literature concerning how the principle of relativity, and more generally the global space-time symmetries, should be understood. On one approach, the significance of space-time symmetries is captured by considering the structure of a theory through transformations on its models, those models consisting of differentiable manifolds endowed with various geometric objects and relations (see Anderson, 1967, and Norton, 1989). According to Brown and Sypel (1995) and Budden (1997), this approach fails to recognise the central importance of effectively isolated subsystems, the empirical significance of symmetries resting on the possibility of transforming such a subsystem (rather than applying the transformation to the entire universe). For further developments in this debate, including applications to local symmetries and to gauge theories, see Kosso (2000), Brading and Brown (2004), Healey (2007), Healey (2009), Greaves and Wallace (2014), Friederich (2015), Rovelli (2014) and Teh (2015, 2016).

The global spacetime invariance principles are intended to be valid for all the laws of nature, for all the processes that unfold in the spacetime. This universal character is not shared by the physical symmetries that were next introduced in physics. Most of these were of an entirely new kind, with no roots in the history of science, and in some cases expressly introduced to describe specific forms of interactions — whence the name “dynamical symmetries” due to Wigner (1967, see especially pp. 15, 17–18, 22–27, 33).

2.1.2 The general theory of relativity

Einstein’s general theory of relativity (GTR) was also constructed using a symmetry principle at its heart: the principle of general covariance. Much ink has been spilled over the significance and role of general covariance in GTR, including by Einstein himself.[10]

For a long time he viewed the principle of general covariance as an extension of the principle of relativity found in both classical mechanics and STR, and this is a view that continues to provoke vigorous debate. Norton (2003) discusses the “Kretschmann objection” to the physical significance of general covariance. On invariance versus covariance, see Anderson (1967), Brown and Brading (2002), and Martin (2003, Section 2.2).

What is clear is that the mere requirement that a theory be generally covariant represents no restriction on the form of the theory; further stipulations must be added, such as the requirement that there be no “absolute objects” (this itself being a problematic notion).

Once some such further requirements are added, however, the principle of general covariance becomes a powerful tool. For a recent review and analysis of this debate, see Pitts (2006).

In Einstein’s hands the principle of general covariance was a crucial postulate in the development of GTR.[11] The diffeomorphism freedom of GTR, i.e., the invariance of the form of the laws under transformations of the coordinates depending smoothly on arbitrary functions of space and time, is a “local” spacetime symmetry, in contrast to the “global” spacetime symmetries of STR (which depend instead on constant parameters).

For a discussion of coordinate-based approaches to the diffeomorphism invariance of General Relativity, see Wallace (forthcoming), and for more on the physical interpretation of this invariance, see Pooley (2017). Such local symmetries are “dynamical” symmetries in Wigner’s sense, since they describe a particular interaction, in this case gravity. As is well known, the spacetime metric in GTR is no longer a “background” field or an “absolute object”, but instead it is a dynamical player, the gravitational field manifesting itself as spacetime curvature.

The extension of the concept of continuous symmetry from “global” symmetries (such as the Galilean group of spacetime transformations) to “local” symmetries is one of the important developments in the concept of symmetry in physics that took place in the twentieth century.

Continuous symmetry 개념을 Galilean group of spacetime transformations 과 같은 전역 대칭 (global symmetries) 에서 국소 대칭(local symmetries)으로 확장한 것은 20세기 물리학 속 symmetry 개념에서 이루어진 중요한 발전 중 하나이다. GTR에서 파생되어, Weyl이 1918년 쓴 "unified theory of gravitation and electromagnetism”는 local symmetires의 아이디어 (see Ryckman, 2003, and Martin, 2003) 를 확장시킨다. 비록 이 이론은 일반적으로 실패한 것으로 여겨지지만, 양자 이론의 문맥에서 성공을 이루는데 주요한 시발점이 된다.

다른 한편으로, Hilbert와 Klein은 중력 이론에서 general covariance의 역할에 대해 상세한 분석을 진행했고, 이 이론에 대해 에너지 보존의 status에 대한 논쟁에 Noether의 도움을 얻었다. 이는 Noether의 유명한 1918년 논문으로 이어졌고, 이 논문에는 두 가지 정리가 담겨있다. 첫 번째 정리는 global symmetries와 conservation laws 간의 연관성을 제시하며, 두 번째 정리는 local symmetries와 관련된 여러 결과를 도출한다. 여기에는 global symmetry group이 해당 이론의 local symmetry group의 subgroup일 때 보존 법칙의 status가 어떻게 달라지는지를 보여주는 내용도 포함된다 (see Brading and Brown, 2003).

2.2 Symmetry and Quantum Mechanics

1920년대 양자역학에서, 대칭상을 활용하기 위해 군론적 표현 방법을 적용한 것은 의심할 여지 없이 20세기 물리학 대칭성 역사에 있어 두 번째 터닝포인트였다. 사실, 대칭성 원리는 양자역학적 문맥에서 가장 효과적으로 작용한다. Wigner, Weyl은 양자역학에서 symmetry group이 양자역학에 끼치는 중요한 관련성을 처음으로 인식하였고, 그 의미에 대해 숙고한 선구자들 중 하나였다. Wigner가 여러 차례 강조 했듯이, 양자이론에서 invariacne princple의 효과성이 증가한 주요 이유 중 하나는 양자 물리학 시스템의 상태 공간 (state space)가 선형적이라는 점이다. 이는 양자 상태를 중첩할 수 있는 가능성과 대응하며, 대칭이 존재할 때 특히 단순한 transformation 성질을 가진 상태를 정의할 수 있게 끔 가능성을 제공한다.

일반적으로, 만약 G를 물리적 시스템을 묘사하는 theory의 symmetry group이라 하면(다시 말해, theory의 동역학 방정식은 G 변환에 대해 불변이다), 이는 시스템의 states가 G 그룹의 어떠한 "representation"에 따라 서로 변화한다는 것이다. 다시 말해, group transfomation은 state space속 states 간의 관계를 나타내는 연산을 통해 수학적으로 기술된다. Roberts는 이를 state space theory에서 "dynamical evolution"과 "spatial translation"과 같은 개념이 의미를 갖기 위한 요구 조건으로 설명하며, 이를 Representation View라고 명시했다.

양자역학의 state space에서, 이러한 오퍼레이션들은 state space에 작용하는 연산자(operator)를 통해 구현되며, 이 연산자는 observables에 해당한다. 물리적 시스템의 모든 state는 basis의 superposition으로 표현될 수 있는데, 여기서 basis는 "irreducible" represenation of the symmetry group에 따라 변환된다. 그러므로 양자역학은 symmetry principle을 적용하는데 있어 특히 우호적인 프레임워크를 제공한다. Observable(state space 에서 이론의 대칭 작용을 나타내며, 그러므로 시스템의 Hamiltonian과 commute함) 은 보존량(conserved quantities)의 역할을 한다. 더욱이, symmetry group의 invariants의 eigenvalue spectra는 irreducible representations of the group을 분류할 수 있도록 하는 라벨을 제공한다.n this fact is grounded the possibility of associating the values of the invariant properties characterizing physical systems with the labels of the irreducible representations of symmetry groups, i.e. of classifying elementary physical systems by studying the irreducible representations of the symmetry groups.

2.3 Permutation symmetry

미시물리학에서 도입된 최초의 non-spatiotemporal symmetry이자, 양자역학에서 group theory 기법으로 처음 다뤄진 대칭은 permuation theory (permutation group의 transformation 변환에 대한 불변성)이었다. 이 대칭은 1926년 W. Heisenberg에 의해 처음 발견 됐으며, 원자 시스템 내에서 동일한 전자들의 구별 불가능성과 연관되어 이산 대칭성(discrete symmetry, 즉 이산 집합의 원소로 이루어진 군에 기반)로 불린다. 이는 소위 quantum statics (Bose-Einstein 과 Fermi-Dirac statistics)의 핵심이며, 특정한 구별 불가능한 양자 입자들(boson, fermion 등)의 ensemble의 통계적 거동을 관장한다. Permutation symmetry principle은, 이러한 ensemble이 이를 구성하는 입자들의 permutation에 대해 invariant하다면, exhcange state는 original state와 동일시 함을 명시한다.

철학적 관점에서 봤을 때, permutation symmetry는 두개의 궁금점을 자아낸다.

1. Permuation symmetry는 한 원자 시스템 속에서 identical particles의 physical indistinguishability 조건으로 간주되며, 이는 quantum domain에서 identity, individuality, indistinguishability 개념의 의미에 대한 논쟁을 불러 일으켰다. "이는 quantum particles들이 개별적인 존재(individuals)가 아니라는 것을 뜻하는가?", "물리적으로 indistinguishable 하지만, 개수로서는 구분 가능한 entity의 존재(problem of identical particles)는 Leibniz’s Principle of the Identity of Indiscernibles이 양자역학에서 위배되는 것으로 간주되어야 함을 시사하는가?" 등의 논쟁이다.

2. 또 다른 관점에서, 이 symmetry principle의 이론적, 그리고 경험적(empirical) status는 무엇인가? 이 법칙은 양자역학의 axiom으로 관주될 수 있는가, 아니면 단순히 경험적으로 정당화된 것으로 받아들여야 하는가? 현재 이 원리는 fermionic, bosonic quantum statistics의 본질을 설명하는데 사용된다. 하지만 permutation symmetry group이 더 다양한 종류의 가능성을 허용함에도, 현재 우리는 bosons과 fermions에 대해서만 알고 있는가? French와 Rickles(2003)는 위 질문들과 관련된 문제들을 제안하고, 이 주제에 대한 새로운 관점은 Saunders(2006)에서 찾을 수 있다. Saunders는 고전 물리학에서의 치환 대칭을 논의하며, 고전 통계를 따르는 구별 불가능한 고전 입자에 대한 주장을 제시한다. 그는 특정 입자 부류의 경우, 양자 통계와 고전 통계 간의 차이를 단순히 구별 불가능성만으로는 완전히 설명할 수 없다고 주장한다. 이와 관련한 더 심도 있는 논의 및 참고문헌으로는 French와 Krause(2006), Ladyman과 Bigaj(2010), Caulton과 Butterfield(2012), 그리고 SEP의 "identity and individuality in quantum theory" 항목을 참조할 수 있다.

2.4 C, P, T

Because of the specific properties of the quantum description, the discrete symmetries of spatial reflection symmetry or parity (P) and time reversal (T) were “rediscovered” in the quantum context, taking on a new significance. Parity was introduced in quantum physics in 1927 in a paper by Wigner, where important spectroscopic results were explained for the first time on the basis of a group-theoretic treatment of permutation, rotation and reflection symmetries. Time reversal invariance appeared in the quantum context, again due to Wigner, in a 1932 paper.[13] To these was added the new quantum particle-antiparticle symmetry or charge conjugation (C). Charge conjugation was introduced in Dirac’s famous 1931 paper “Quantized singularities in the electromagnetic field”.

The discrete symmetries C, P and T are connected by the so-called CPT theorem, first proved by Lüders and Pauli in the early 1950s, which states that the combination of C, P, and T is a general symmetry of Lorentz-invariant physical laws. For philosophical reflections on the meaning and grounding of the CPT theorem see Wallace (2009) and Greaves (2010). Greaves (2010) in particular argues that the quantization of the classical T operator leads to a CT operator, using this in support of Feynman’s famous suggestion that time reversal has the effect of transforming particles into anti-particles. Roberts (2022, 2.8) argues against this claim, by maintaining that the Feynman’s approach is an artifact that does not accurately capture the description of temporal symmetry. A discussion of the proofs of the theorem, both within the standard quantum field theory framework and the axiomatic field theory framework, is provided in Greaves and Thomas (2014).

The laws governing gravity, electromagnetism, and the strong interaction are invariant with respect to C, P and T independently. However, in 1956 T. D. Lee and C. N. Yang pointed out that β-decay, governed by the weak interaction, had not yet been tested for invariance under P. Soon afterwards C. S. Wu and her colleagues performed an experiment showing that the weak interaction violates parity. Nevertheless, β-decay respects the combination of C and P as a symmetry. In 1964, however, CP was found to be violated in weak interaction by Cronin and Fitch in an experiment involving K-mesons. This implied, in virtue of the CPT theorem, the violation of T-symmetry as well; since then, there have been direct observations of T-symmetry violation, as reported by e.g., CPLEAR Collaboration (1998). For a careful analysis of the underlying assumptions working in the assessment of T-violation see Roberts (2015) and Ashtekar (2015). The philosophical puzzle is how we can come to know this time asymmetry, given that we cannot truly ‘reverse’ time. Roberts proposes three templates for explaining how this is possible, drawing in particular on a version of Curie’s principle. By contrast, Gołosz (2016) argues that this asymmetry should be understood as applying to a physical process, rather than to time itself. Roberts (2022, Chapter 7) responds that, on the Representation View, the asymmetry does indeed apply to time itself.

This debate stems from the more basic question of what time reversal symmetry actually means. Some have argued that the textbook time reversal operator T does not really represent the reversal of time alone, due to its unusual properties like conjugation of the wavefunction and the reversal of magnetic fields, including Albert (2000), Callender (2000), Lopez (2021, 2023). Others maintain, instead, that the T operator is indeed the right definition of time reversal, including Earman (2002), Malament (2004) and Roberts (2017). An ecumenical response is given by Roberts (2022, Chapter 2), arguing that these two approaches can be viewed as the two sides of the same coin, corresponding, respectively, to a reversal of time translations on spacetime and to the representation of that reversal on state space. A response in a similar spirit for electromagnetism is given by Struyve (forthcoming).

The existence of parity violation in our fundamental laws has led to a new chapter in an old philosophical debate concerning chiral or handed objects and the nature of space. A description of a left hand and one of a right hand will not differ so long as no appeal is made to anything beyond the relevant hand. Yet left and right hands do differ — a left-handed glove will not fit on a right hand. For a brief period, Kant saw in this reason to prefer a substantivalist account of space over a relational one, the difference between left and right hands lying in their relation to absolute space. Regardless of whether this substantivalist solution succeeds, there remains the challenge to the relationalist of accounting for the difference between what Kant called “incongruent counterparts” — objects which are the mirror-image of one another and yet cannot be made to coincide by any rigid motion. The relationalist may respond by denying that there is any intrinsic difference between a left and a right hand, and that the incongruence is to be accounted for in terms of the relations between the two hands (if a universe was created with only one hand in it, it would be neither left nor right, but the second hand to be created would be either incongruent or congruent with it). This response becomes problematic in the face of parity violation, where one possible experimental outcome is much more likely than its mirror-image. Since the two possible outcomes don’t differ intrinsically, how should we account for the imbalance? This issue continues to be discussed in the context of the substantivalist-relationalist debate. For further details see Pooley (2003) and Saunders (2007).

2.5 Gauge symmetry

The starting point for the idea of continuous internal symmetries was the interpretation of the presence of particles with (approximately) the same value of mass as the components (states) of a single physical system, connected to each other by the transformations of an underlying symmetry group. This idea emerged by analogy with what happened in the case of permutation symmetry, and was in fact due to Heisenberg (the discoverer of permutation symmetry), who in a 1932 paper introduced the SU(2) symmetry connecting the proton and the neutron (interpreted as the two states of a single system). This symmetry was further studied by Wigner, who in 1937 introduced the term isotopic spin (later contracted to isospin). The various internal symmetries are invariances under phase transformations of the quantum states and are described in terms of the unitary groups SU(N). The term “gauge” is sometimes used for all continuous internal symmetries, and is sometimes reserved for the local versions (these being at the core of the Standard Model for elementary particles).[14]

The phase of the quantum wavefunction encodes internal degrees of freedom. With the requirement that a theory be invariant under local gauge transformations involving the phase of the wavefunction, Weyl’s ideas of 1918 found a successful home in quantum theory (see O’Raifeartaigh, 1997). Weyl’s new 1929 theory was a theory of electromagnetism coupled to matter. The history of gauge theory is surveyed briefly by Martin (2003), who highlights various issues surrounding gauge symmetry, in particular the status of the so-called “gauge principle”, first proposed by Weyl. A recent account of the conceptual structure of the gauge argument is provided by Gomes, Roberts and Butterfield (2022). The main steps in development of gauge theory are the Yang and Mills non-Abelian gauge theory of 1954, and the problems and solutions associated with the successful development of gauge theories for the short-range weak and strong interactions. In subsequent years, we have come to understand that “local” gauge symmetries are not just a feature of electromagnetism, but also of the majority of our best theories, including various gravitational theories and models within condensed matter physics.

The main philosophical questions raised by gauge theory all hinge upon how we should understand the relationship between mathematics and physics. There are two broad categories of discussion. The first concerns the gauge principle, already mentioned, and the issue here is the extent to which the requirement that we write our theories in locally-symmetric form enables us to derive new physics. The analysis concerns listing what premises constitute the gauge principle, examining the status of these premises and what motivation might be given for them, determining precisely what can be obtained on the basis of these premises, and what more needs to be added in order to arrive at a (successful) physical theory. For discussion of this argument, see Teller (2000), Martin (2003) and Gomes et al (2022). In Teh (forthcoming), the case is made that the debate surrounding the gauge principle is related to the debate about the difference between substantive and non-substantive forms of “general covariance”, a term that is used to pick out the “local gauge (diffeomorphism) symmetry” of general relativity.

The second category concerns the question of which quantities in a gauge theory represent the “physically real” properties. This question arises acutely in gauge theories because of the apparent failure of determinism if certain redundancies of description are not taken into account in formulating the initial value problem. The problem was first encountered in GTR (which in this respect is a gauge theory), and for further details the best place to begin is with the literature on Einstein’s “hole argument” (see Earman and Norton, 1987; Earman, 1989, Chapter 9; and more recently Norton, 1993; Rynasiewicz, 1999; Saunders, 2002; Pooley, 2021; and the references therein). In practice, we find that only gauge-invariant quantities are observables, and this seems to rescue us. However, this is not the end of the story. The other canonical example is the Aharanov-Bohm effect, and we can use this to illustrate the interpretational problem associated with gauge theories, sometimes characterized as a dilemma: failure of determinism or action-at-a-distance (see Healey, 2001). Restoring determinism depends on only gauge-invariant quantities being taken as representing “physically real” quantities, but accepting this solution apparently leaves us with some form of non-locality between causes and effects.

Furthermore, we face the question of how to understand the role of the non-gauge-invariant quantities appearing in the theory, and the problem of how to interpret what M. Redhead calls “surplus structure” (see Redhead, 2003).

For an early discussion of “surplus structure” in the philosophical literature, see Belot (1998) and Nounou (2003). More recently, the topic of “surplus structure” in gauge theories has been discussed in Weatherall (2016), Murray, Nguyen & Teh (2020), Gomes (2021), Murgueitio-Ramirez and Teh (forthcoming), Dougherty (2020) and Wallace (2022): the more recent discussion draws interesting connections between the discussion of gauge theory and cutting edge topics in theoretical physics, e.g. the application of category theory to physics, and the use of the covariant phase space formalism.

For an approach to these questions using the theory of constrained Hamiltonian systems, see also Earman (2003b), Castellani (2003, 2004), and Pitts (2014). For an intuitive characterization of gauge symmetry, one that is more general than the Lagrangian and Hamiltonian formulations of theories in which gauge symmetry is usually expressed, see Belot (2008).

How best to interpret gauge theories is an open issue in the philosophy of physics. Healey (2007) discusses the conceptual foundations of gauge theories, arguing in favour of a non-separable holonomy interpretation of classical Yang-Mills gauge theories of fundamental interactions. Catren (2008) tackles the ontological implications of Yang-Mills theory by means of the fiber bundle formalism. Useful references are the Metascience review symposium on Healey (2007) (Rickles, Smeenk, Lyre and Healey, 2009), and the “Synopsis and Discussion” of the workshop “Philosophy of Gauge Theory,” Center for Philosophy of Science, University of Pittsburgh, 18–19 April 2009 (available online).

A recent, detailed overview and discussion of the various issues raised by the question of the status (both in physics and in philosophy of physics) of gauge symmetries is provided in Berghofer, François, Friederich, Gomes, Hetzroni, Maas, and Sondenheimer (2023).

2.6 Dualities

Thus far, we have been discussing symmetries which act on the space of states of a physical theory. In recent years, much discussion within physics and philosophy has centered on certain kinds of symmetries that act on the space of theories. When such symmetries are interpreted as realizing an “equivalence” (which sense is itself something that requires philosophical work to explicate) between two theories, the theories are typically said to be related by a “duality symmetry” (and if we are speaking of “symmetry” in the strict sense of an automorphism, then such dualities are called “self-dualities”).

The dualities playing a central role in contemporary physics are of various types: dualities between quantum field theories (like the generalized electric-magnetic duality), between string theories (such as T-duality and S-duality), and between physical descriptions which are, respectively, a quantum field theory and a string theory, as in the case of gauge/gravity dualities. Historically, the first relevant dualities to be used in physics were electric-magnetic duality, momentum-position duality (via Fourier transform) – wave-particle duality in the QM context – and the Kramers-Wannier duality of the two-dimensional Ising model in statistical physics. On the various types of dualities and their significance in physics, see in particular Castellani and Rickles (2017). An introductory review of gauge/gravity duality, emphasising the conceptual aspects, is De Haro, Mayerson, and Butterfield (2016).

In general, because dualities are transformations between theories, their implications are more radical than those of symmetry transformations. While symmetries are mapping between the solutions of the same theory, different theoretical descriptions can have very different interpretations in terms of objects, properties, degrees of freedom, and spacetime frameworks. Thus, dualities naturally offer a new and interesting viewpoint on many traditional issues in the philosophy of science, such as a) reduction, emergence, and fundamentality (for a discussion of duality and emergence see, for example, Rickles, 2013, and Castellani and De Haro, 2020); b) theoretical equivalence and underdetermination (see Read and Møller-Nielsen, 2020, on the import of duality for these issues); and c) realism versus anti-realism (Le Bihan and Read, 2018, provide a survey of the landscape of possible ontological interpretations of duality-related theories; a specific example, is the case of the “particle democracy” related to weak/strong dualities, as discussed in Castellani, 2017). A comprehensive discussion of the many issues related to the physics and philosophy of dualities is De Haro and Butterfield (forthcoming).

Within the philosophical literature, work relating symmetries to dualities generally responds to one of the following three questions:

What is an appropriate formal framework for understanding the inter-theoretic relationship realized by duality symmetries? Within the context of quantum gauge/gravity duality, a rudimentary framework for doing so was sketched in De Haro, Teh, and Butterfield (2017) and then further developed in De Haro and Butterfield (2017) for the case of bosonization dualities. A comprehensive, detailed survey is provided in De Haro and Butterfield (2018). And within the far simpler context of classical mechanics, Teh and Tsementzis (2017) explore the use of the quintessential symmetry of classical phase space (viz. symplectomorphisms) to relate Hamiltonian and Lagrangian theories, and proceed to discuss how can use this framework as a toy model for thinking about more sophisticated dualities.
What is the relationship between the local symmetry of a theory and the symmetries of its dual theory? One reason that this question is pressing is that (as we mentioned in Section 2.5), local symmetries are typically regard as “surplus” or “redundant” structure of a theory; thus one might hope to be able to construct a dual theory that does not contain such surplus structure. Indeed, it was conjectured in Polchinski and Horowitz (2009) that for gravity/gauge duality, the local symmetries (i.e. the diffeomorphisms) of the bulk theory will always be “invisible” from the perspective of its dual boundary theory. However, De Haro, Teh, and Butterfield (2017) argue that this conjecture is not true in full generality: there is a special class of diffeomorphisms which do not vanish in the boundary theory, but instead correspond to conformal transformations of the boundary CFT.
Should duality symmetries themselves by understood as a certain kind of “gauge symmetry”, thus explaining why we should treat dual theories as “physically equivalent”? On this question, Read (2016) undertakes the task of using the “hole argument” as a lens with which to compare string-theoretic dualities with (intra-theory) gauge symmetries.

3. SYMMETRY ARGUMENTS

Consider the following cases.

Buridan’s ass: situated between what are, for him, two completely equivalent bundles of hay, he has no reason to choose the one located to his left over the one located to his right, and so he is not able to choose and dies of starvation.
Archimedes’s equilibrium law for the balance: if equal weights are hung at equal distances along the arms of a balance, then it will remain in equilibrium since there is no reason for it to rotate one way or the other about the balance point.
Anaximander’s argument for the immobility of the Earth as reported by Aristotle: the Earth remains at rest since, being at the centre of the spherical cosmos (and in the same relation to the boundary of the cosmos in every direction), there is no reason why it should move in one direction rather than another.

What do they have in common?

First, these can all be understood as examples of the application of the Leibnizean Principle of Sufficient Reason (PSR): if there is no sufficient reason for one thing to happen instead of another, the principle says that nothing happens (the initial situation does not change). But there is something more that the above cases have in common: in each of them PSR is applied on the grounds that the initial situation has a given symmetry: in the first two cases, bilateral symmetry; in the third, rotational symmetry. The symmetry of the initial situation implies the complete equivalence between the existing alternatives (the left bundle of hay with respect to the right one, and so on). If the alternatives are completely equivalent, then there is no sufficient reason for choosing between them and the initial situation remains unchanged.

만약 특정 사건이 다른 사건보다 발생할 충분한 이유가 없다면, 해당 원리에 따르면 아무 일도 일어나지 않으며 초기 상황이 변하지 않습니다. 그러나 위 사례들에는 또 다른 공통점이 있습니다. 각 사례에서 충분한 이유 원리(Principle of Sufficient Reason, PSR)가 적용되는 근거는 초기 상황이 특정 **대칭성(symmetry)**을 가진다는 것입니다.

첫 번째와 두 번째 사례에서는 **좌우 대칭성(bilateral symmetry)**이 적용되고,
세 번째 사례에서는 **회전 대칭성(rotational symmetry)**이 적용됩니다.

초기 상황의 대칭성은 주어진 대안들(예: 왼쪽 건초 더미와 오른쪽 건초 더미 등) 사이에 완전한 동등성이 존재함을 의미합니다. 이러한 대안들이 완전히 동등하다면, 그 중 하나를 선택할 충분한 이유가 없기 때문에 초기 상황이 그대로 유지됩니다.

이 논리는 물리학에서 자발적 대칭 깨짐(spontaneous symmetry breaking) 개념과도 연관될 수 있습니다. 초기 상태의 대칭성은 선택의 여지를 없애며, 외부 요인 없이 변화가 일어나지 않음을 설명합니다.

Arguments of the above kind — that is, arguments leading to definite conclusions on the basis of an initial symmetry of the situation plus PSR — have been used in science since antiquity (as Anaximander’s argument testifies). The form they most frequently take is the following: a situation with a certain symmetry evolves in such a way that, in the absence of an asymmetric cause, the initial symmetry is preserved. In other words, a breaking of the initial symmetry cannot happen without a reason, or an asymmetry cannot originate spontaneously. Van Fraassen (1989) devotes a chapter to considering the way these kinds of symmetry arguments can be used in general problem-solving.

Historically, the first explicit formulation of this kind of argument in terms of symmetry is due to the physicist Pierre Curie towards the end of nineteenth century. Curie was led to reflect on the question of the relationship between physical properties and symmetry properties of a physical system by his studies on the thermal, electric and magnetic properties of crystals, these properties being directly related to the structure, and hence the symmetry, of the crystals studied. More precisely, the question he addressed was the following: in a given physical medium (for example, a crystalline medium) having specified symmetry properties, which physical phenomena (for example, which electric and magnetic phenomena) are allowed to happen? His conclusions, systematically presented in his 1894 work “Sur la symétrie dans les phénomènes physiques”, can be synthesized as follows:

A phenomenon can exist in a medium possessing its characteristic symmetry or that of one of its subgroups. What is needed for its occurrence (i.e. for something rather than nothing to happen) is not the presence, but rather the absence, of certain symmetries: “Asymmetry is what creates a phenomenon”.
The symmetry elements of the causes must be found in their effects, but the converse is not true; that is, the effects can be more symmetric than the causes.

Conclusion (a) clearly indicates that Curie recognized the important function played by the concept of symmetry breaking in physics (he was indeed one of the first to recognize it). Conclusion (b) is what is usually called “Curie’s principle” in the literature, although notice that (a) and (b) are not independent of one another.

In order for Curie’s principle to be applicable, various conditions need to be satisfied: the causal connection must be valid, the cause and effect must be well-defined, and the symmetries of both the cause and the effect must also be well-defined (this involves both the physical and the geometrical properties of the physical systems considered). Curie’s principle then furnishes a necessary condition for given phenomena to happen: only those phenomena can happen that are compatible with the symmetry conditions established by the principle.

Curie’s principle has thus an important methodological function: on the one side, it furnishes a kind of selection rule (given an initial situation with a specified symmetry, only certain phenomena are allowed to happen); on the other side, it offers a falsification criterion for physical theories (a violation of Curie’s principle may indicate that something is wrong in the physical description).[15]

Such applications of Curie’s principle depend, of course, on our accepting its validity, and this is something that has been questioned in the literature, especially in relation to spontaneous symmetry breaking (see below, next section). Different proposals have been offered for justifying the principle. We have presented it here as an example of symmetry considerations based on Leibniz’s PSR, while Curie himself seems to have regarded it as a form of the causality principle. In current literature, it has become standard to understand the principle as following from the invariance properties of deterministic physical laws. According to this “received view”, as first formulated in Chalmers (1970) and then developed in more recent literature (Ismael 1997, Belot 2003, Earman 2004), Curie’s principle is expressed in terms of the relationship between the symmetries of earlier and later states of a system, and the laws connecting these states. In fact, this is a misrepresentation of Curie’s original principle: Curie’s focus was on the case of co-existing, functionally related features of a system’s state, rather than temporally ordered cause and effect pairs. For a discussion on such questions as to whether there is more that one “Curie’s principle” and, independently of how it is formulated, what are the aspects that make it so scientically fruitful, see Castellani and Ismael (2016). As regards the status of the principle, Norton (2016) argues that it is a “truism”, on the grounds that whether the principle succeeds or fails depends on how one chooses to attach causal labels to the scientific description. For Roberts (2013), Curie’s principle fails when the symmetry is time reversal. Roberts (2016) claims that the truth of the principle is contingent on special physical facts, and attributes to its failure an important role in the detection of parity violation and CP violation (on this point, see also Roberts, 2015, and Ashtekar, 2015).

4. SYMMETRY BREAKING

A symmetry can be exact, approximate, or broken. Exact means unconditionally valid; approximate means valid under certain conditions; broken can mean different things, depending on the object considered and its context.

The study of symmetry breaking also goes back to Pierre Curie. According to Curie, symmetry breaking has the following role: for the occurrence of a phenomenon in a medium, the original symmetry group of the medium must be lowered (broken, in today’s terminology) to the symmetry group of the phenomenon (or to a subgroup of the phenomenon’s symmetry group) by the action of some cause. In this sense symmetry breaking is what “creates the phenomenon”. Generally, the breaking of a certain symmetry does not imply that no symmetry is present, but rather that the situation where this symmetry is broken is characterized by a lower symmetry than the original one. In group-theoretic terms, this means that the initial symmetry group is broken to one of its subgroups. It is therefore possible to describe symmetry breaking in terms of relations between transformation groups, in particular between a group (the unbroken symmetry group) and its subgroup(s). As is clearly illustrated in the 1992 volume by I. Stewart and M. Golubitsky, starting from this point of view a general theory of symmetry breaking can be developed by tackling such questions as “which subgroups can occur?”, “when does a given subgroup occur?”

Symmetry breaking was first explicitly studied in physics with respect to physical objects and phenomena. This is not surprising, since the theory of symmetry originated with the visible symmetry properties of familiar spatial figures and every day objects. However, it is with respect to the laws that symmetry breaking has acquired special significance in physics. There are two different types of symmetry breaking of the laws: “explicit” and “spontaneous”, the case of spontaneous symmetry breaking being the more interesting from a physical as well as a philosophical point of view.

4.1 Explicit symmetry breaking

Explicit symmetry breaking indicates a situation where the dynamical equations are not manifestly invariant under the symmetry group considered. This means, in the Lagrangian (Hamiltonian) formulation, that the Lagrangian (Hamiltonian) of the system contains one or more terms explicitly breaking the symmetry. Such terms can have different origins:

(a) Symmetry-breaking terms may be introduced into the theory by hand on the basis of theoretical/experimental results, as in the case of the quantum field theory of the weak interactions, which is expressly constructed in a way that manifestly violates mirror symmetry or parity. The underlying result, in this case, is parity non-conservation in the case of the weak interaction, first predicted in the famous (Nobel-prize winning) 1956 paper by T. D. Lee and C.N. Yang.

(b) Symmetry-breaking terms may appear in the theory because of quantum-mechanical effects. One reason for the presence of such terms — known as “anomalies” — is that in passing from the classical to the quantum level, because of possible operator ordering ambiguities for composite quantities such as Noether charges and currents, it may be that the classical symmetry algebra (generated through the Poisson bracket structure) is no longer realized in terms of the commutation relations of the Noether charges. Moreover, the use of a “regulator” (or “cut-off”) required in the renormalization procedure to achieve actual calculations may itself be a source of anomalies. It may violate a symmetry of the theory, and traces of this symmetry breaking may remain even after the regulator is removed at the end of the calculations. Historically, the first example of an anomaly arising from renormalization is the so-called chiral anomaly, that is the anomaly violating the chiral symmetry of the strong interaction (see Weinberg, 1996, Chapter 22).

(c) Finally, symmetry-breaking terms may appear because of non-renormalizable effects. Physicists now have good reasons for viewing current renormalizable field theories as effective field theories, that is low-energy approximations to a deeper theory (each effective theory explicitly referring only to those particles that are of importance at the range of energies considered). The effects of non-renormalizable interactions (due to the heavy particles not included in the theory) are small and can therefore be ignored at the low-energy regime. It may then happen that the coarse-grained description thus obtained possesses more symmetries than the deeper theory. That is, the effective Lagrangian obeys symmetries that are not symmetries of the underlying theory. These “accidental” symmetries, as Weinberg has called them, may then be violated by the non-renormalizable terms arising from higher mass scales and suppressed in the effective Lagrangian (see Weinberg, 1995, pp. 529–531).

4.2 Spontaneous symmetry breaking

Spontaneous symmetry breaking (SSB) occurs in a situation where, given a symmetry of the equations of motion, solutions exist which are not invariant under the action of this symmetry without any explicit asymmetric input (whence the attribute “spontaneous”).[16] A situation of this type can be first illustrated by means of simple cases taken from classical physics. Consider for example the case of a linear vertical stick with a compression force applied on the top and directed along its axis. The physical description is obviously invariant for all rotations around this axis. As long as the applied force is mild enough, the stick does not bend and the equilibrium configuration (the lowest energy configuration) is invariant under this symmetry. When the force reaches a critical value, the symmetric equilibrium configuration becomes unstable and an infinite number of equivalent lowest energy stable states appear, which are no longer rotationally symmetric but are related to each other by a rotation. The actual breaking of the symmetry may then easily occur by effect of a (however small) external asymmetric cause, and the stick bends until it reaches one of the infinite possible stable asymmetric equilibrium configurations.[17] In substance, what happens in the above kind of situation is the following: when some parameter reaches a critical value, the lowest energy solution respecting the symmetry of the theory ceases to be stable under small perturbations and new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are asymmetric but are all related through the action of the symmetry transformations. In other words, there is a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole set of which maintains the symmetry of the theory.

In quantum physics SSB actually does not occur in the case of finite systems: tunnelling takes place between the various degenerate states, and the true lowest energy state or “ground state” turns out to be a unique linear superposition of the degenerate states. In fact, SSB is applicable only to infinite systems — many-body systems (such as ferromagnets, superfluids and superconductors) and fields — the alternative degenerate ground states being all orthogonal to each other in the infinite volume limit and therefore separated by a “superselection rule” (see for example Weinberg, 1996, pp. 164–165).

Historically, the concept of SSB first emerged in condensed matter physics. The prototype case is the 1928 Heisenberg theory of the ferromagnet as an infinite array of spin 1/2 magnetic dipoles, with spin-spin interactions between nearest neighbours such that neighbouring dipoles tend to align. Although the theory is rotationally invariant, below the critical Curie temperature Tc the actual ground state of the ferromagnet has the spin all aligned in some particular direction (i.e. a magnetization pointing in that direction), thus not respecting the rotational symmetry. What happens is that below Tc there exists an infinitely degenerate set of ground states, in each of which the spins are all aligned in a given direction. A complete set of quantum states can be built upon each ground state. We thus have many different “possible worlds” (sets of solutions to the same equations), each one built on one of the possible orthogonal (in the infinite volume limit) ground states. To use a famous image by S. Coleman, a little man living inside one of these possible asymmetric worlds would have a hard time detecting the rotational symmetry of the laws of nature (all his experiments being under the effect of the background magnetic field). The symmetry is still there — the Hamiltonian being rotationally invariant — but “hidden” to the little man. Besides, there would be no way for the little man to detect directly that the ground state of his world is part of an infinitely degenerate multiplet. To go from one ground state of the infinite ferromagnet to another would require changing the directions of an infinite number of dipoles, an impossible task for the finite little man (Coleman, 1975, pp. 141–142). As said, in the infinite volume limit all ground states are separated by a superselection rule. Ruetsche (2006) discusses symmetry breaking and ferromagnetism from the algebraic perspective. Liu and Emch (2005) address the interpretative problems of explaining SSB in nonrelativistic quantum statistical mechanics. Fraser (2016) discusses SBB in finite systems, arguing against the indispensability of the thermodynamic limit in the characterization of SSB in statistical mechanics.

The same picture can be generalized to quantum field theory (QFT), the ground state becoming the vacuum state, and the role of the little man being played by ourselves. This means that there may exist symmetries of the laws of nature which are not manifest to us because the physical world in which we live is built on a vacuum state which is not invariant under them. In other words, the physical world of our experience can appear to us very asymmetric, but this does not necessarily mean that this asymmetry belongs to the fundamental laws of nature. SSB offers a key for understanding (and utilizing) this physical possiblity.

The concept of SSB was transferred from condensed matter physics to QFT in the early 1960s, thanks especially to works by Y. Nambu and G. Jona-Lasinio. Jona-Lasinio (2003) offers a first-hand account of how the idea of SSB was introduced and formalized in particle physics on the grounds of an analogy with the breaking of (electromagnetic) gauge symmetry in the 1957 theory of superconductivity by J. Bardeen, L. N. Cooper and J. R. Schrieffer (the so-called BCS theory). The application of SSB to particle physics in the 1960s and successive years led to profound physical consequences and played a fundamental role in the edification of the current Standard Model of elementary particles. In particular, let us mention the following main results that obtain in the case of the spontaneous breaking of a continous internal symmetry in QFT.

Goldstone theorem. In the case of a global continuous symmetry, massless bosons (known as “Goldstone bosons”) appear with the spontaneous breakdown of the symmetry according to a theorem first stated by J. Goldstone in 1960. The presence of these massless bosons, first seen as a serious problem since no particles of the sort had been observed in the context considered, was in fact the basis for the solution — by means of the so-called Higgs mechanism (see the next point) — of another similar problem, that is the fact that the 1954 Yang-Mills theory of non-Abelian gauge fields predicted unobservable massless particles, the gauge bosons.

Higgs mechanism. According to a “mechanism” established in a general way in 1964 independently by (i) P. Higgs, (ii) R. Brout and F. Englert, and (iii) G. S. Guralnik, C. R. Hagen and T. W. B. Kibble, in the case that the internal symmetry is promoted to a local one, the Goldstone bosons “disappear” and the gauge bosons acquire a mass. The Goldstone bosons are “eaten up” to give mass to the gauge bosons, and this happens without (explicitly) breaking the gauge invariance of the theory. Note that this mechanism for the mass generation for the gauge fields is also what ensures the renormalizability of theories involving massive gauge fields (such as the Glashow-Weinberg-Salam electroweak theory developed in the second half of the 1960s), as first generally demonstrated by M. Veltman and G. ’t Hooft in the early 1970s. The Higgs mechanism it at the center of a lively debate among philosophers of physics: see, for example, Smeenk (2006), Lyre (2008), Struyve (2011), Friederich (2013). For a historical-philosophical analysis, see also Borrelli (2012). A detailed discussion of the foundational and interpretational issues raised by the Higgs mechanism is provided in Berghofer, François, Friederich, Gomes, Hetzroni, Maas, and Sondenheimer (2023, 3.3).

Dynamical symmetry breaking (DSB). In such theories as the unified model of electroweak interactions, the SSB responsible (via the Higgs mechanism) for the masses of the gauge vector bosons is because of the symmetry-violating vacuum expectation values of scalar fields (the so-called Higgs fields) introduced ad hoc in the theory. For different reasons — first of all, the initially ad hoc character of these scalar fields for which there was no experimental evidence until the results obtained in July 2012 at the LHC — some attention has been drawn to the possibility that the Higgs fields could be phenomenological rather than fundamental, that is bound states resulting from a specified dynamical mechanism. SSB realized in this way has been called “DSB”.[18]

Symmetry breaking raises a number of philosophical issues. Some of them relate only to the breaking of specific types of symmetries, such as the issue of the significance of parity violation for the problem of the nature of space (see Section 2.4, above). Others, for example the connection between symmetry breaking and observability, are particular aspects of the general issue concerning the status and significance of physical symmetries, but in the case of SSB they take on a stronger force: what is the epistemological status of a theory based on “hidden” symmetries and SSB? Given that what we directly observe — the physical situation, the phenomenon — is asymmetric, what is the evidence for the “underlying” symmetry? On this point, see for example Morrison (2003) and Kosso (2000). In the absence of direct empirical evidence, the above question then becomes whether and how far the predictive and explanatory power of theories based on SSB provides good reasons for believing in the existence of the hidden symmetries. Finally, there are issues raised by the motivation for, and role of, SSB. See for example Earman (2003a), using the algebraic formulation of QFT to explain SSB; for further philosophical discussions on SBB in QFT in the algebraic approach, see Ruetsche (2011), Fraser (2012), and references therein. Landsman (2013) discusses the issue whether SBB in infinite quantum systems can be seen as an example of asymptotic emergence in physics. Roberts (2022, 4.5.4) argues that SBB provides an example of spacetime symmetries that fail to be dynamical symmetries, contra a famous principle of Earman (1989, 3.4).

SSB allows symmetric theories to describe asymmetric reality. In short, SSB provides a way of understanding the complexity of nature without renouncing fundamental symmetries. But why should we prefer symmetric to asymmetric fundamental laws? In other words, why assume that an observed asymmetry requires a cause, which can be an explicit breaking of the symmetry of the laws, asymmetric initial conditions, or SSB? Note that this assumption is very similar to the one expressed by Curie in his famous 1894 paper. Curie’s principle (the symmetries of the causes must be found in the effects; or, equivalently, the asymmetries of the effects must be found in the causes), when extended to include the case of SSB, is equivalent to a methodological principle according to which an asymmetry of the phenomena must come from the breaking (explicit or spontaneous) of the symmetry of the fundamental laws. What the real nature of this principle is remains an open issue, at the centre of a developing debate (see Section 3, above).

Finally, let us mention the argument that is sometimes made in the literature that SSB implies that Curie’s principle is violated because a symmetry is broken “spontaneously”, that is without the presence of any asymmetric cause. Now it is true that SSB indicates a situation where solutions exist that are not invariant under the symmetry of the law (dynamical equation) without any explicit breaking of this symmetry. But, as we have seen, the symmetry of the “cause” is not lost, it is conserved in the ensemble of the solutions (the whole “effect”).[19]

5. GENERAL PHILOSOPHICAL QUESTIONS

Much of the recent philosophical literature on symmetries in physics discusses specific symmetries and the intepretational questions they lead to. The rich variety of symmetries in modern physics means that questions concerning the status and significance of symmetries in physics in general are not easily addressed. However, something can be said in more general terms and we offer a few remarks in that direction here, starting with the main roles that symmetry plays in physics.

One of the most important roles played by symmetry is that of classification — for example, the classification of crystals using their remarkable and varied symmetry properties. In contemporary physics, the best example of this role of symmetry is the classification of elementary particles by means of the irreducible representations of the fundamental physical symmetry groups, a result first obtained by Wigner in his famous paper of 1939 on the unitary representations of the inhomogeneous Lorentz group. When a symmetry classification includes all the necessary properties for characterizing a given type of physical object (for example, all necessary quantum numbers for characterizing a given type of particle), we have the possibility of defining types of entities on the basis of their transformation properties. This has led philosophers of science to explore a structuralist approach to the entities of modern physics, in particular a group-theoretical account of objects (see the contributions in Castellani, 1998, Part II). The relation between symmetry, group and structure and its exploitation in structuralist approaches, especially in the framework of structural realism, is a much debated issue. See, for example, Roberts (2011), French (2014) and Caulton (2015), and the SEP entry Structural Realism.

Symmetries also have a normative role, being used as constraints on physical theories. The requirement of invariance with respect to a transformation group imposes severe restrictions on the form that a theory may take, limiting the types of quantities that may appear in the theory as well as the form of its fundamental equations. A famous case is Einstein’s use of general covariance when searching for his gravitational equations.

The group-theoretical treatment of physical symmetries, with the resulting possibility of unifying different types of symmetries by means of a unification of the corresponding transformation groups, has provided the technical resources for symmetry to play a powerful role in theoretical unification. This is best illustrated by the current dominant research programme in theoretical physics aimed at arriving at a unified description of all the fundamental forces of nature (gravitational, weak, electromagnetic and strong) in terms of underlying local symmetry groups.

It is often said that many physical phenomena can be explained as (more or less direct) consequences of symmetry principles or symmetry arguments. In the case of symmetry principles, the explanatory role of symmetries arises from their place in the hierarchy of the structure of physical theory, which in turn derives from their generality. As Wigner (1967, pp. 28ff) describes the hierarchy, symmetries are seen as properties of the laws. Symmetries may be used to explain (i) the form of the laws, and (ii) the occurrence (or non-occurrence) of certain events (this latter in a manner analogous to the way in which the laws explain why certain events occur and not others). In the case of symmetry arguments, we may, for example, appeal to Curie’s principle to explain the occurrence of certain phenomena on the basis of the symmetries (or asymmetries) of the situation, as discussed in section 3, above. Furthermore, insofar as explanatory power may be derived from unification, the unifying role of symmetries also results in an explanatory role.

From these different roles we can draw some preliminary conclusions about the status of symmetries. It is immediately apparent that symmetries have an important heuristic function, indicating a strong methodological status. Is this methodological power connected to an ontological or epistemological status for symmetries?

According to an ontological viewpoint, symmetries are seen as a substantial part of the physical world: the symmetries of theories represent properties existing in nature, or characterize the structure of the physical world. It might be claimed that the ontological status of symmetries provides the reason for the methodological success of symmetries in physics. A concrete example is the use of symmetries to predict the existence of new particles. This can happen via the classificatory role, on the grounds of vacant places in symmetry classification schemes, as in the famous case of the 1962 prediction of the particle Omega- in the context of the hadronic classification scheme known as the “Eightfold Way”. (See Bangu, 2008, for a critical analysis of the reasoning leading to this prediction.) Or, as in more recent cases, via the unificatory role: the paradigmatic example is the prediction of the W and Z particles (experimentally found in 1983) in the context of the Glashow-Weinberg-Salam gauge theory proposed in 1967 for the unification of the weak and electromagnetic interactions. These impressive cases of the prediction of new phenomena might then be used to argue for an ontological status for symmetries, via an inference to the best explanation.

Another reason for attributing symmetries to nature is the so-called geometrical interpretation of spatiotemporal symmetries, according to which the spatiotemporal symmetries of physical laws are interpreted as symmetries of spacetime itself, the “geometrical structure” of the physical world. Moreover, this way of seeing symmetries can be extended to non-external symmetries, by considering them as properties of other kinds of spaces, usually known as “internal spaces”. The question of exactly what a realist would be committed to on such a view of internal spaces remains open, and an interesting topic for discussion.

One approach to investigating the limits of an ontological stance with respect to symmetries would be to investigate their empirical or observational status: can the symmetries in question be directly observed? We first have to address what it means for a symmetry to be observable, and indeed whether all symmetries have the same observational status. Kosso (2000) arrives at the conclusion that there are important differences in the empirical status of the different kinds of symmetries. In particular, while global continuous symmetries can be directly observed — via such experiments as the Galilean ship experiment — a local continuous symmetry can have only indirect empirical evidence. Brading and Brown (2004) argue for a different interpretation of Kosso’s examples,[20] but agree with Kosso’s assessment that local symmetry transformations cannot exhibit an analogue of the Galileo’s Ship Experiment. Against this view, Greaves and Wallace (2014) and Teh (2016) have recently argued that when suitably understood, local symmetries can indeed have direct empirical significance (their argument turns on the possibility of using asymptotically non-trivial local symmetries to construct Galileo’s Ship scenarios for certain systems). In particular, Teh (2016) argues that this phenomenon is responsible for the empirically significant symmetries of topological soliton solutions. In a similar spirit, Gomes (2021) argues that a certain class of gauge transformations are on a par with Galileo’s Ship, under special conditions, and that those conditions imply a certain kind of holism in the sense that the state of the Universe is not uniquely determined by the state of its isolated subsystems. On the other hand, Friederich (2015) contends that on one plausible axiomatization of the schema introduced by Greaves and Wallace, it is possible to deduce that local symmetries do not have direct empirical significance. This debate was recently expanded by Roberts (2022, 4.3) to include discrete symmetry transformations like time reversal. Roberts argues that these symmetries have direct empirical significance, but at a “higher-order” level, that derives from the direct empirical significance of spacetime translations.

The direct observational status of the familiar global spacetime symmetries leads us to an epistemological aspect of symmetries. According to Wigner, the spatiotemporal invariance principles play the role of a prerequisite for the very possibility of discovering the laws of nature: “if the correlations between events changed from day to day, and would be different for different points of space, it would be impossible to discover them” (Wigner, 1967, p. 29). For Wigner, this conception of symmetry principles is essentially related to our ignorance (if we could directly know all the laws of nature, we would not need to use symmetry principles in our search for them). Others, on the contrary, have arrived at a view according to which symmetry principles function as “transcendental principles” in the Kantian sense (see for instance Mainzer, 1996). It should be noted in this regard that Wigner’s starting point, as quoted above, does not imply exact symmetries — all that is needed epistemologically is that the global symmetries hold approximately, for suitable spatiotemporal regions, such that there is sufficient stability and regularity in the events for the laws of nature to be discovered.

There is another reason why symmetries might be seen as being primarily epistemological. As we have mentioned, there is a close connection between the notions of symmetry and equivalence, and this leads also to a notion of irrelevance: the equivalence of space points (translational symmetry), for example, may be understood in the sense of the irrelevance of an absolute position to the physical description. There are two ways that one might interpret the epistemological significance of this: on the one hand, we might say that symmetries are associated with unavoidable redundancy in our descriptions of the world, while on the other hand we might maintain that symmetries indicate a limitation of our epistemic access — there are certain properties of objects, such as their absolute positions, that are not observable. The view that symmetries are connected with the presence of non-observable quantities in the physical description, with the corollary that the empirical violation of a symmetry is intended in the sense that what was thought to be a non-observable turns out to be actually an observable, was particularly defended by one of the discover of parity violation, T.D. Lee (see section 2.4). According to Lee (1981), “the root of all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities” (p. 178). See on this (and, more generally, on the relation between symmetry, equivalence and irrelevance) Castellani (2003). Dasgupta (2016) defends an epistemic interpretation of symmetry on a similar basis as Lee, that he calls the “symmetry to reality inference”. In contrast, Roberts (2022, 4.4) argues that the symmetry to reality inference is not generally valid.

Finally, we would like to mention an aspect of symmetry that might very naturally be used to support either an ontological or an epistemological account. It is widely agreed that there is a close connection between symmetry and objectivity, the starting point once again being provided by spacetime symmetries: the laws by means of which we describe the evolution of physical systems have an objective validity because they are the same for all observers. The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), “objectivity means invariance with respect to the group of automorphisms [of space-time]”.[21] Debs and Redhead (2007) label as “invariantism” the view that “invariance under a specified group of automorphisms is both a necessary and sufficient condition for objectivity” (p. 60). They point out (p. 73, and see also p. 66) that there is a natural connection between “invariantism” and structural realism.

Growing interest, recently, in the metaphysics of physics includes interest in symmetries. Baker (2010) offers an accessible introduction, and Livanios (2010), connecting discussions of symmetries to dispositions and essences, is an example of this work.

To conclude: symmetries in physics offer many interpretational possibilities, and how to understand the status and significance of physical symmetries clearly presents a challenge to both physicists and philosophers. A recent, comprehensive survey of the philosophy of symmetry is Teh forthcoming.

대칭성은 exact하거나 appoximate하거나 아니면 broken 총 세개의 가능성이 있다. 이를 하나씩 살펴본다

1. Exact

이 경우는 조건 없이 유효하다는 의미이다.

2. Approximate

이 경우는 특정 조건에서 유효하다는 의미이다.

3. Broken

이 경우는 고려하는 물체나 문맥에 따라 뜻하는 바가 다를 수도 있다.

대칭성붕괴에 대한 연구는 Pierre Curie까지 거슬러 올라간다. Curie에 따르면, 대칭성붕괴는 다음의 역할을 갖는다.

' 물체에서 나타는 현상의 발현에 대해서, the original symmetry group of the medium must be broken to the symmetry group of the phenomenon by the action of some cause.

이 관점에서 본다면, 대칭성 붕괴는 어떠한 현상을 만들어내는 무언가이다.

일반적으로, 특정 대칭성의 붕괴가 '대칭성이 존재하지 않다'는 것을 뜻하지는 않는다. 되려 대칭성이 붕괴된 그 상황은, 원래것에 비해 더 낮은 대칭성을 갖는다고 특정 지을 수 있다.

Group-theoretic 용어에서, 위의 문장은 초기의 대칭성 그룹이 여러 서브 그룹중 하나의 서브그룹으로 붕괴된다는 것을 의미한다.

그러므로 '대칭성붕괴'라는 용어는 transformation group들 사이의 관계의 관점에서 설명이 가능하다.

특히 between a group (the unbroken symmetry group) and its subgroup(s).

이는 1992년 I. Stewart and M. Golubitsky저의 책에서 분명하게 설명되어 있음을 알 수 있다.

책의 관점에서 시작하여, 대칭성붕괴에대한 일반적인 general theory는 '어떤 subgroup이 나타날 수 있는가?', '언제 특정 subgroup이 나타나는가?' 하는 질문들에 태클을 걸며 발전할 수 있었다

대칭성붕괴는 물리학에서 처음 직접적으로 연구되기 시작했으며 물리적인 물체와 현상에 대한 연구가 시발점이었다.

이는 놀라운것이 아닌게, 대칭성 이론은 잘 알려진 공간적 도형과 일상에서 볼 수 있는 물체 등 '가시적인 대칭성 특성'에서 유래했기 때문이다.

하지만 대칭성붕괴가 물리학에서 특별한 상징성을 얻게 된것은 'laws'과 관련이 있다.

대칭성붕괴에는 크게 두가지 laws가 존재하는데, 바로 'explicit', 그리고 'spontaneous'이다.

이중 자발적 대칭성 붕괴(Spontaneous Symmetry Breaking)의 경우가 더 흥미롭다고 말 할 수 있다. 물리적으로, 그리고 철학적 관점에서도!.

4.1. Explicit Symmetry Breaking

Explicit Symmettry Breaking은 dynamical equation이 고려하는 symmetry group에서 manifestly invariant 하지 않은 상황을 나타낸다.

이는 다시 말해, 라그랑지안(혹은 헤밀토니안) formulation에서, 시스템의 라그랑지안 (혹은 헤밀토니안)이 대칭성을 explicitly 붕괴시키는 항을 한개 이상 갖고 있다는 의미이다.

이때 이 항들은 서로 다른 근원을 가질 수 있다.

(A) Symmetry-breaking 항은 이론적, 실험적 결과에 의해 추가 될 수 있다. 약한 상호작용의 양자장이론(QFT, Quantum Field Theory)이 이 경우에 해당할 것이다

(B) 다른 한편으로 Symmetry-breaking 항은 양자역학적 효과 때문에 추가 될 수 있다. 보통 "anomalies"라 불리는 이 항이 존재하는 이유중 하나를 살펴본다.

고전역학에서 양자역학으로 넘어갈 때, because of possible operator ordering ambiguities for composite quantities such as Noether charges and currents, it may be that the classical symmetry algebra (generated through the Poisson bracket structure) is no longer realized in terms of the commutation relations of the Noether charges. Moreover, the use of a “regulator” (or “cut-off”) required in the renormalization procedure to achieve actual calculations may itself be a source of anomalies. It may violate a symmetry of the theory, and traces of this symmetry breaking may remain even after the regulator is removed at the end of the calculations. Historically, the first example of an anomaly arising from renormalization is the so-called chiral anomaly, that is the anomaly violating the chiral symmetry of the strong interaction (see Weinberg, 1996, Chapter 22).
(c) Finally, symmetry-breaking terms may appear because of non-renormalizable effects. Physicists now have good reasons for viewing current renormalizable field theories as effective field theories, that is low-energy approximations to a deeper theory (each effective theory explicitly referring only to those particles that are of importance at the range of energies considered). The effects of non-renormalizable interactions (due to the heavy particles not included in the theory) are small and can therefore be ignored at the low-energy regime. It may then happen that the coarse-grained description thus obtained possesses more symmetries than the deeper theory. That is, the effective Lagrangian obeys symmetries that are not symmetries of the underlying theory. These “accidental” symmetries, as Weinberg has called them, may then be violated by the non-renormalizable terms arising from higher mass scales and suppressed in the effective Lagrangian (see Weinberg, 1995, pp. 529–531).

4.2. 자발적 대칭성 붕괴 (Spontaneous Symmetry Breaking)

자발적 대칭성 붕괴 (Spontaneous Symmetry Breaking, 이하 SSB)는 occurs in a situation where, given a symmetry of the equations of motion, solutions exist which are not invariant under the action of this symmetry without any explicit asymmetric input (whence the attribute “spontaneous”).[16]

SSB에 해당하는 상황은 고전역학에서도 찾아 볼 수 있다.

Linear vertical stick이 서있을 때 수직으로 외력이 가해진다고 생각해본다. 물리적으로 이를 묘사할 때 축을 중심으로 한 회전에 대해서는 모든 방향에 대해 invariant하다.

힘이 너무 강하지 않는 한 stick은 휘어지지 않고 equilibrium configuration (the lowest energy configuration)은 이 대칭성에서 invariant하다.

만약 힘이 임계 값에 도달한다면, the symmetric equilibrium configuration은 불안정해지고, 수많은 equivalent lowest energy stable states가 나타난다. 이 states들은 더 이상 회전 대칭성을 가지진 않지만, 회전을 통해 서로 연관을 가지게 된다.

이 경우 실제 대칭성의 붕괴는 매우 작은 외부의 asymmetric 요인에 의해서라도 쉽게 나타날 것이고, stick은 수많은 stable asymmetric equilibrium configuration 중 한개에 도달할 때 까지 휘어질 것이다.

In substance, what happens in the above kind of situation is the following: when some parameter reaches a critical value, the lowest energy solution respecting the symmetry of the theory ceases to be stable under small perturbations and new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are asymmetric but are all related through the action of the symmetry transformations. In other words, there is a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole set of which maintains the symmetry of the theory.
In quantum physics SSB actually does not occur in the case of finite systems: tunnelling takes place between the various degenerate states, and the true lowest energy state or “ground state” turns out to be a unique linear superposition of the degenerate states. In fact, SSB is applicable only to infinite systems — many-body systems (such as ferromagnets, superfluids and superconductors) and fields — the alternative degenerate ground states being all orthogonal to each other in the infinite volume limit and therefore separated by a “superselection rule” (see for example Weinberg, 1996, pp. 164–165).

역사적으로 봤을때, SSB 개념은 응집물질물리학에서 처음 부상했다.

프로토타입 케이스는 1928년의 아이젠버그의 강자성체 이론 일것이다. 해당 이론은 무한한 spin1/2 자기모멘트의 배열에 대한 것으로 해당 자기모멘트들은 가장 가까운 스핀 사이의 스핀-스핀 상호작용 (spin-spin interaction)에 의해, 이웃하는 자기모멘트와 평행 정렬하려는 경향을 가진다.

비록 이론이 회전 변환에 대해 불변이지만, 큐리온도 $T_{C}$ 밑에서 실제 강자성체의 ground state는 특정 방향(자화를 생각하면 된다)을 향해 정렬하게 되며 rotational symmetry를 만족하지 못한다.

임계값 $T_{C}$ 밑에서는 무한한 degenerated된 바닥 상태가 존재하며, 그 상태속의 전자 스핀은 모두 한 방향을 향하고 있다. 양자 상태의 complete set은 각 ground state에 만들어질 수 있다.

그러므로 우리는 수많은 다른 "가능한 세상"(sets of solutions to the same equations)를 가질 수 있으며, 각각의 세상은 possible orthogonal ground states중 하나에 지어질 수 있다.

S. Coleman에 의해 만들어진 유명한 예를 살펴보자.
Possible asymmetric worlds 중 하나에 살아가는 소형 남자는 rotational symmetry of the laws of nature를 찾기 매우 힘들 것이다. 마치 그가 행하는 모든 실험이 자기장이 걸려 있는 상태에서 실험하는 것과 같을 것이다. 물론 대칭은 확실히 존재한다. 다만, 그 소형 남자에게는 감춰져 있을 뿐이다. 더욱이, 그 소형 남자가 살아가는 세상의 ground sate가 수많은 degenerate multiplet 중 하나라는 것을 직접적으로 발견할 깃은 더더욱 없다. 무한한 강자성체 속 한개의 ground state에서 또 다른 ground state(또 다른 세계)로 넘어가는 것은 수 많은 dipole의 방향을 바꾸는 것을 필요로 한다. 이는 유한한 소형 남자에게는 불가능한 일이다.(Coleman, 1975, pp. 141-142)

말했듯이, 무한한 부피 경계 속에서는 모든 ground states는 superselection rule에 의해 분리되어 있다. Ruetsche는 2006년에 symmetry breaking과 ferromagnetism을 대수학적 관점에서 논의하였다. 이에 앞서 2005년에는 Liu와 Emch가 비상대론적 양자 통계 물리학에서의 SSB를 설명하는 해석적 문제에 대해서 다루었다. 10년이 지난 후 Fraser는 2016년에 infinite system에서의 SBB 를 논의 하였으며, 통계 역학에서 SSB의 characterization에 대한 열역학적 limit의 필수불가결함에 반기를 들었다.

똑같은 문제는 QFT에서도 일반화 될 수 있다. ground state는 vacuum state로, 소형남자의 역할은 우리가 스스로 맡게 된다. 이 말은 즉슨, 자연법칙의 대칭성이 존재할 수 있으면서도, 우리에게는 발현되지 않을 수도 있는데 이는 우리가 살아가고 있는 물리적 세상이 대칭성들에 대해 불변하지 않은 vacuum state를 기반으로 만들어 졌기 때문이다. 다른 말로, 우리가 경험하는 물리적 세상은 우리에게 매우 asymmetric 하게 느껴질 수도 있다는 것이자. 하지만 이로 인해 asymmetry가 자연 법칙의 fundamental에 속한다는 뜻은 아니다. SSB는 이 물리적 가능성을 이해하고 응용할 수 있는 키를 제공한다.

SSB의 개념은 응집물질 물리학에서 이른 1960년대 쯔음에 QFT로 넘어갔다. 이는 Y.Bambu, 그리고 G. HJona- Lasinio의 공이 매우 컸다. Jona-Lasinio는 2003년에 직접 SSB

The concept of SSB was transferred from condensed matter physics to QFT in the early 1960s, thanks especially to works by Y. Nambu and G. Jona- Lasinio.

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Spinor(도른자)와 Spin

대칭성과 대칭성 붕괴 (Symmetry and Symmetry Breaking)

0. INTRODUCTION

1. THE CONCEPT OF SYMMETRY