43:
100:
1581:
419:
1635:. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Currently LHC is preparing for a run which tests supersymmetry.
385:. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by transformations that change
2597:
1273:
is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps
1808:
gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e.
1550:
and the resulting low-entropy state in the "future". Since we perceive the "past" ("future") as having lower (higher) entropy than the present, the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past, and vice
1626:
A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the
Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between
1697:
SO(3). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is SO(3). Any rotation preserves distances on the surface of the ball. The set of all
Lorentz transformations form a group called the
369:; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for
1555:
These symmetries are near-symmetries because each is broken in the present-day universe. However, the
Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called
2739:
1693:). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the
3289:
250:. Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges.
2440:
2956:
1649:
Generalized symmetries encompass a number of recently recognized generalizations of the concept of a global symmetry. These include higher form symmetries, higher group symmetries, non-invertible symmetries, and subsystem symmetries.
1082:
above the Earth's surface. Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time
322:
The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if
3329:
In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields.
1140:
and describe those situations where a property of the system does not change with a continuous change in location. For example, the temperature in a room may be independent of where the thermometer is located in the
3325:
under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the
Universe may have and finding the invariants to construct field theories as models.
2434:
3024:
3125:
If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form:
2849:
148:
of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by
1217:: These are spatio-temporal symmetries which generalise Poincaré transformations to include other conformal one-to-one transformations on the space-time coordinates. Lengths are not invariant under
2866:
is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field:
1240:
associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.
1465:
3120:
1542:
is reversed. T-symmetry is counterintuitive (the future and the past are not symmetrical) but explained by the fact that the
Standard Model describes local properties, not global ones like
2603:
165:
These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as
1411:
trajectory through the air, whether the recording is played normally or in reverse. Thus, position is symmetric with respect to the instant that the object is at its maximum height.
3131:
200:
Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example,
1319:
1362:
1510:
has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.
1407:. This may be illustrated by recording the motion of an object thrown up vertically (neglecting air resistance) and then playing it back. The object will follow the same
2319:
2287:
836:
2592:{\displaystyle \delta \psi ^{\alpha }(x)=h^{\mu }(x)\partial _{\mu }\psi ^{\alpha }(x)+\partial _{\mu }h_{\nu }(x)\sigma _{\mu \nu }^{\alpha \beta }\psi ^{\beta }(x)}
3420:
Cordova, Clay; Dumitrescu, Thomas; Intriligator, Kenneth; Shao, Shu-Heng (2022). "Snowmass White Paper: Generalized
Symmetries in Quantum Field Theory and Beyond".
3047:
2771:
2351:
2251:
2872:
884:
1405:
1382:
1467:
and indicate an invariance property of a system when the coordinates are 'inverted'. Stated another way, these are symmetries between a certain object and its
172:
Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is described in
204:
may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is
3060:
Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind:
889:
2209:
Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle
879:
874:
1251:
structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of
3394:
3389:
3053:. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of
694:
3354:
1291:: Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation,
958:
841:
3776:
3754:
3650:
3608:
3587:
3562:
3532:
3485:
1526:(parity symmetry), a universe where everything is mirrored along the three physical axes. This excludes weak interactions as demonstrated by
1208:
3836:
989:
3399:
64:
2363:
1161:. The latter are represented by square matrices with determinant â1 and consist of a proper rotation combined with a spatial reflection (
3798:
2217:
of two of these infinitesimal transformations is equivalent to a third infinitesimal transformation of the same kind hence they form a
3811:
Click on link to
Chapter 6: Symmetry, Invariance, and Conservation for a simplified, step-by-step introduction to symmetry in physics.
211:
Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit
2962:
3730:
3632:
3507:
3464:
86:
851:
1907:
1669:
1322:
2779:
1709:
Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the
123:
is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some
3946:
1072:
389:
as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.
246:
from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius
3379:
2354:
1763:
1665:
846:
826:
166:
1422:
315:
and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the
3931:
3066:
1107:
791:
699:
1165:). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article
3384:
381:
The two examples of rotational symmetry described above â spherical and cylindrical â are each instances of
188:
of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in
365:
is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of
57:
51:
3936:
1834:
1744:
831:
1157:. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit
1564:, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of
1218:
124:
31:
2734:{\displaystyle \delta A_{\mu }(x)=h^{\nu }(x)\partial _{\nu }A_{\mu }(x)+A_{\nu }(x)\partial _{\mu }h^{\nu }(x)}
68:
3941:
3829:
1883:
1872:
1810:
1694:
982:
466:
3284:{\displaystyle h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }+Dx_{\mu }+K^{\mu }|x|^{2}-2K^{\nu }x_{\nu }x_{\mu },}
3369:
1928:
1771:
1740:
1644:
1539:
357:
is one that keeps a property invariant for a transformation that is applied simultaneously at all points of
239:
3572:
Brading, K.; Castellani, E. (2007). "Symmetries and
Invariances in Classical Physics". In Butterfield, J.;
1479:: These are represented by a composition of a translation and a reflection. These symmetries occur in some
3691:
1859:
1814:
1186:
786:
749:
717:
704:
234:
For example, an electric field due to an electrically charged wire of infinite length is said to exhibit
1896:
1877:
1853:
1756:
1071:
in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have
818:
486:
1294:
1328:
3683:
2859:
2085:
2050:
2033:
1805:
1795:
1244:
1237:
446:
436:
398:
145:
3696:
3900:
3822:
3450:
3364:
3322:
1848:
1829:
1675:
1178:
1145:
975:
963:
804:
634:
386:
382:
235:
1820:
The following table summarizes some fundamental symmetries and the associated conserved quantity.
3709:
3673:
3421:
3374:
3314:
1980:
1783:
1190:
1182:
735:
725:
231:
when discussing observed physical symmetry; this can be applied to symmetries in forces as well.
212:
189:
185:
173:
3951:
3772:
3750:
3726:
3646:
3604:
3583:
3558:
3528:
3503:
3495:
3481:
3460:
3340:
3318:
2295:
2263:
2102:
2016:
1962:
1801:
1703:
1264:
1167:
1154:
914:
799:
762:
652:
181:
3744:
3640:
3624:
Laws, Symmetry, and
Symmetry Breaking: Invariance, Conservations Principles, and Objectivity.
3577:
3859:
3740:
3701:
3598:
3552:
3475:
3454:
3359:
3344:
2951:{\displaystyle \delta \psi ^{\alpha }(x)=\lambda (x).\tau ^{\alpha \beta }\psi ^{\beta }(x)}
2183:
2090:
1843:
1775:
1767:
1752:
1569:
1523:
1507:
1475:
1415:
1162:
1031:
934:
614:
606:
598:
590:
582:
561:
551:
541:
531:
515:
496:
456:
216:
3032:
99:
3349:
2747:
2327:
2227:
2210:
2038:
1710:
1565:
1527:
1484:
1233:
1226:
1150:
919:
672:
657:
428:
120:
3310:
1762:
Also, the reduction by symmetry of the energy functional under the action by a group and
1207:. They may be defined on any smooth manifold, but find many applications in the study of
3687:
1387:
3802:
3662:"Reflections on the four facets of symmetry: how physics exemplifies rational thinking"
2254:
2196:
1736:
1732:
1659:
1503:
1367:
939:
924:
757:
662:
276:
158:
140:
104:
3925:
3764:
3446:
2156:
2055:
1699:
1621:
1497:
1248:
647:
476:
3723:
Angular
Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems
3713:
3622:
3887:
3882:
3514:
Chapter 12 is a gentle introduction to symmetry, invariance, and conservation laws.
2322:
2290:
2258:
2132:
2006:
1748:
1725:
1679:
1561:
1557:
1518:
1468:
1229:
944:
929:
730:
712:
642:
370:
153:
3705:
1674:
The transformations describing physical symmetries typically form a mathematical
3793:
3618:
3573:
3518:
2218:
2168:
2070:
1985:
1970:
1158:
1054:
1035:: A physical system may have the same features over a certain interval of time Î
770:
686:
410:
201:
108:
3661:
3864:
3854:
3522:
2214:
2024:
1997:
1779:
1535:
1514:
1287:
1252:
909:
775:
667:
17:
1813:, and temporal translation symmetry (i.e. homogeneity of time) gives rise to
1181:, i.e. they are isometries of Minkowski space. They are studied primarily in
3905:
3050:
2021:
1690:
1204:
406:
366:
358:
177:
149:
2744:
Without gravity only the Poincaré symmetries are preserved which restricts
1800:
The symmetry properties of a physical system are intimately related to the
1111:: These spatial symmetries are represented by transformations of the form
1039:; this is expressed mathematically as invariance under the transformation
3877:
3814:
2144:
1933:
1901:
1747:
group. (Roughly speaking, the symmetries of the SU(3) group describe the
1547:
1408:
1200:
1018:, involving only the spatial geometry associated with a physical system;
135:
3794:
The Feynman Lectures on Physics Vol. I Ch. 52: Symmetry in Physical Laws
1568:
in the universe. CP violation is a fruitful area of current research in
1517:(charge symmetry), a universe where every particle is replaced with its
1103:, the particle's total gravitational potential energy will be preserved.
2114:
1632:
1546:. To properly reverse the direction of time, one would have to put the
1543:
1480:
866:
3808:
3642:
Symmetries of Nature: A Handbook for Philosophy of Nature and Science
3600:
Objectivity, Invariance, and Convention: Symmetry in Physical Science
1766:
of transformations of symmetric groups appear to elucidate topics in
418:
3426:
2429:{\displaystyle \delta \phi (x)=h^{\mu }(x)\partial _{\mu }\phi (x)}
2224:
A general coordinate transformation described as the general field
1177:: These are spatio-temporal symmetries which preserve distances in
3678:
2177:
2123:
1628:
1247:
which are those spacetime symmetries that preserve the underlying
1007:
253:
In Newton's theory of mechanics, given two bodies, each with mass
98:
2079:
2064:
2047:
2030:
1011:
279:
of the system (as calculated from an observer at the origin) is
3818:
3019:{\displaystyle \delta A_{\mu }(x)=\partial _{\mu }\lambda (x),}
27:
Feature of a system that is preserved under some transformation
3500:
Timeless Reality: Symmetry, Simplicity, and Multiple Universes
1574:
1225:
Mathematically, spacetime symmetries are usually described by
169:
and can, in addition, be exploited to simplify many problems.
36:
3477:
Deep Down Things: The Breathtaking Beauty of Particle Physics
1221:
but there is a cross-ratio on four points that is invariant.
3302:
generating special conformal transformations. For example,
1199:: These are spatio-temporal symmetries which preserve the
1185:. Those isometries that leave the origin fixed are called
3524:
Fearful Symmetry: The search for beauty in modern physics
2844:{\displaystyle h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }}
208:
under a shift in an observer's position within the room.
1731:
and the symmetries natural to such a theory are called
1595:
1419:: These are represented by transformations of the form
1685:
Continuous symmetries are specified mathematically by
3134:
3069:
3035:
2965:
2875:
2782:
2750:
2606:
2443:
2366:
2330:
2298:
2266:
2230:
1425:
1390:
1370:
1331:
1297:
3317:
does not although other theories of gravity such as
1682:
is an important area of mathematics for physicists.
219:
of the sphere will preserve how the sphere "looks".
2862:(giving the Lorentz and rotational symmetries) and
1460:{\displaystyle {\vec {r}}\,\rightarrow -{\vec {r}}}
3283:
3115:{\displaystyle \delta \phi (x)=\Omega (x)\phi (x)}
3114:
3041:
3018:
2950:
2843:
2765:
2733:
2591:
2428:
2345:
2313:
2281:
2245:
1459:
1399:
1376:
1356:
1313:
3554:Symmetries in Physics: Philosophical Reflections
837:Representation theory of semisimple Lie algebras
1538:(time reversal symmetry), a universe where the
257:, starting at the origin and moving along the
130:A family of particular transformations may be
3830:
1243:Some of the most important vector fields are
1149:: These spatial symmetries are classified as
983:
261:-axis in opposite directions, one with speed
8:
3527:(2nd ed.). Princeton University Press.
1026:, involving changes in both space and time.
1006:are symmetries involving transformations of
227:The above ideas lead to the useful idea of
152:while discrete symmetries are described by
3837:
3823:
3815:
990:
976:
875:Particle physics and representation theory
520:
417:
402:
3695:
3677:
3425:
3395:Standard Model (mathematical formulation)
3390:List of mathematical topics in relativity
3321:do. The 'action' of a field theory is an
3272:
3262:
3252:
3236:
3231:
3222:
3216:
3203:
3187:
3174:
3161:
3139:
3133:
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3034:
2995:
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2933:
2920:
2883:
2874:
2835:
2822:
2809:
2787:
2781:
2749:
2716:
2706:
2687:
2665:
2655:
2636:
2614:
2605:
2574:
2561:
2553:
2534:
2524:
2502:
2492:
2473:
2451:
2442:
2408:
2389:
2365:
2329:
2297:
2265:
2229:
1446:
1445:
1438:
1427:
1426:
1424:
1389:
1369:
1343:
1342:
1335:
1330:
1301:
1296:
87:Learn how and when to remove this message
1822:
1483:and in some planar symmetries, known as
345:Symmetries may be broadly classified as
50:This article includes a list of general
3412:
1189:and give rise to the symmetry known as
842:Representations of classical Lie groups
574:
523:
405:
3809:Pedagogic Aids to Quantum Field Theory
3298:generating scale transformations and
1209:exact solutions in general relativity
1022:, involving only changes in time; or
1014:. These may be further classified as
176:by a group of transformations of the
7:
3551:Brading, K.; Castellani, E. (2003).
2257:) has the infinitesimal effect on a
1809:homogeneity of space) gives rise to
695:Lie groupâLie algebra correspondence
3799:Stanford Encyclopedia of Philosophy
3631:Address to the 2002 meeting of the
3582:. North Holland. pp. 1331â68.
3456:Symmetry and the Beautiful Universe
1720:A type of physical theory based on
184:. Another important example is the
3805:"âby K. Brading and E. Castellani.
3480:. Johns Hopkins University Press.
3088:
2992:
2703:
2652:
2521:
2489:
2405:
56:it lacks sufficient corresponding
25:
3633:Philosophy of Science Association
2353:that can be expressed (using the
1811:conservation of (linear) momentum
1755:and the U(1) group describes the
1314:{\displaystyle t\,\rightarrow -t}
1751:, the SU(2) group describes the
1739:, used to describe three of the
1702:(this may be generalised to the
1670:Symmetries in general relativity
1654:Mathematics of physical symmetry
1579:
1357:{\displaystyle F\,=m{\ddot {r}}}
1325:still holds if, in the equation
41:
3400:WheelerâFeynman absorber theory
3313:theory has this symmetry while
3049:are generators of a particular
3597:Debs, T.; Redhead, M. (2007).
3557:. Cambridge University Press.
3232:
3223:
3151:
3145:
3109:
3103:
3097:
3091:
3082:
3076:
3010:
3004:
2985:
2979:
2945:
2939:
2910:
2904:
2895:
2889:
2799:
2793:
2760:
2754:
2728:
2722:
2699:
2693:
2677:
2671:
2648:
2642:
2626:
2620:
2586:
2580:
2546:
2540:
2514:
2508:
2485:
2479:
2463:
2457:
2423:
2417:
2401:
2395:
2379:
2373:
2340:
2334:
2308:
2302:
2276:
2270:
2240:
2234:
1790:Conservation laws and symmetry
1451:
1439:
1432:
1302:
1073:gravitational potential energy
890:Galilean group representations
885:Poincaré group representations
1:
3721:Thompson, William J. (1994).
3380:Harmonic coordinate condition
3355:Covariance and contravariance
2355:Einstein summation convention
1764:spontaneous symmetry breaking
1666:Symmetry in quantum mechanics
1323:Newton's second law of motion
1078:when suspended from a height
880:Lorentz group representations
847:Theorem of the highest weight
3603:. Harvard University Press.
1804:characterizing that system.
3749:. Oxford University Press.
3666:European Physical Journal H
3579:Philosophy of Physic Part B
3385:Inertial frame of reference
2165:SU(3) gauge transformation
2111:SU(2) gauge transformation
1590:may have misleading content
3968:
3769:Symmetries and Reflections
3706:10.1140/epjh/e2013-40018-4
1793:
1735:. Gauge symmetries in the
1663:
1657:
1642:
1619:
1495:
1262:
1024:spatio-temporal symmetries
832:Lie algebra representation
396:
29:
3896:
3873:
3850:
1219:inversion transformations
1215:Inversion transformations
268:and the other with speed
32:Symmetry (disambiguation)
2314:{\displaystyle \psi (x)}
2282:{\displaystyle \phi (x)}
2056:lepton generation number
1967:P, coordinate inversion
1741:fundamental interactions
1695:special orthogonal group
1175:Poincaré transformations
827:Lie group representation
3370:Principle of covariance
2153:SU(3) "winding number"
1724:symmetries is called a
1645:Non-invertible symmetry
1187:Lorentz transformations
852:BorelâWeilâBott theorem
240:electric field strength
215:. A rotation about any
196:As a kind of invariance
111:showing symmetry labels
71:more precise citations.
3845:C, P, and T symmetries
3285:
3116:
3043:
3020:
2952:
2845:
2767:
2735:
2593:
2430:
2347:
2315:
2283:
2247:
1815:conservation of energy
1639:Generalized symmetries
1461:
1401:
1378:
1358:
1315:
750:Semisimple Lie algebra
705:Adjoint representation
112:
3947:Differential geometry
3286:
3117:
3044:
3042:{\displaystyle \tau }
3021:
2953:
2846:
2768:
2736:
2594:
2431:
2348:
2316:
2284:
2248:
2129:gauge transformation
2067:gauge transformation
1757:electromagnetic force
1462:
1402:
1379:
1359:
1316:
1274:usually being called
1245:Killing vector fields
1238:local diffeomorphisms
1197:Projective symmetries
819:Representation theory
167:group representations
102:
3660:Mouchet, A. (2013).
3639:Mainzer, K. (1996).
3502:. Prometheus Books.
3459:. Prometheus Books.
3132:
3067:
3033:
2963:
2873:
2858:is an antisymmetric
2780:
2766:{\displaystyle h(x)}
2748:
2604:
2441:
2364:
2346:{\displaystyle A(x)}
2328:
2296:
2264:
2246:{\displaystyle h(x)}
2228:
2086:gauge transformation
2051:gauge transformation
2034:gauge transformation
2011:product of parities
1873:translation in space
1841:Proper orthochronous
1745:SU(3) Ă SU(2) Ă U(1)
1743:, are based on the
1485:wallpaper symmetries
1423:
1388:
1368:
1329:
1295:
1004:spacetime symmetries
399:Spacetime symmetries
242:at a given distance
236:cylindrical symmetry
30:For other uses, see
3932:Concepts in physics
3688:2013EPJH...38..661M
3474:Schumm, B. (2004).
3365:Galilean invariance
2773:to be of the form:
2569:
1849:translation in time
1596:clarify the content
1179:Minkowski spacetime
1108:Spatial translation
1020:temporal symmetries
964:Table of Lie groups
805:Compact Lie algebra
383:continuous symmetry
223:Invariance in force
3911:Symmetry (physics)
3375:General covariance
3315:general relativity
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1981:charge conjugation
1835:Conserved quantity
1784:physical cosmology
1770:(for example, the
1457:
1400:{\displaystyle -t}
1397:
1374:
1354:
1311:
1191:Lorentz covariance
1183:special relativity
1155:improper rotations
1016:spatial symmetries
736:Affine Lie algebra
726:Simple Lie algebra
467:Special orthogonal
337:are interchanged.
213:spherical symmetry
190:general relativity
174:special relativity
113:
3937:Conservation laws
3919:
3918:
3778:978-0-262-73021-1
3756:978-0-19-151999-4
3746:Laws and symmetry
3652:978-3-11-088693-1
3610:978-0-674-03413-6
3589:978-0-08-046665-1
3564:978-1-139-44202-2
3544:Technical readers
3534:978-0-691-00946-9
3487:978-0-8018-7971-5
3341:Conserved current
3319:conformal gravity
2253:(also known as a
2202:
2201:
2103:electroweak force
2017:Internal symmetry
1994:T, time reversal
1963:Discrete symmetry
1897:rotation in space
1806:Noether's theorem
1802:conservation laws
1796:Noether's theorem
1687:continuous groups
1613:
1612:
1540:direction of time
1454:
1435:
1416:Spatial inversion
1377:{\displaystyle t}
1351:
1271:discrete symmetry
1265:Discrete symmetry
1236:. The underlying
1168:Rotation symmetry
1137:
1128:
1119:
1000:
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800:Split Lie algebra
763:Cartan subalgebra
625:
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516:Simple Lie groups
97:
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16:(Redirected from
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3771:. M.I.T. Press.
3760:
3741:Van Fraassen, B.
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2091:weak hypercharge
1908:angular momentum
1844:Lorentz symmetry
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1776:electromagnetism
1768:particle physics
1753:weak interaction
1733:gauge symmetries
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1508:particle physics
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792:Complexification
635:Other Lie groups
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429:Classical groups
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341:Local and global
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138:of a circle) or
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1528:Chien-Shiung Wu
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3788:External links
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2197:Standard Model
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2813:
2810:
2806:
2802:
2796:
2788:
2784:
2776:
2775:
2774:
2757:
2751:
2725:
2717:
2713:
2707:
2696:
2688:
2684:
2680:
2674:
2666:
2662:
2656:
2645:
2637:
2633:
2629:
2623:
2615:
2611:
2607:
2600:
2583:
2575:
2571:
2565:
2562:
2557:
2554:
2550:
2543:
2535:
2531:
2525:
2517:
2511:
2503:
2499:
2493:
2482:
2474:
2470:
2466:
2460:
2452:
2448:
2444:
2437:
2420:
2414:
2409:
2398:
2390:
2386:
2382:
2376:
2370:
2367:
2360:
2359:
2358:
2356:
2337:
2331:
2324:
2305:
2299:
2292:
2273:
2267:
2260:
2256:
2237:
2231:
2222:
2220:
2216:
2212:
2204:
2198:
2195:
2191:
2189:
2188:
2185:
2182:
2179:
2176:
2174:
2173:
2170:
2167:
2164:
2162:
2161:
2158:
2157:baryon number
2155:
2152:
2150:
2149:
2146:
2143:
2140:
2138:
2137:
2134:
2131:
2125:
2122:
2120:
2119:
2116:
2113:
2110:
2108:
2107:
2104:
2101:
2098:
2096:
2095:
2092:
2089:
2087:
2081:
2078:
2076:
2075:
2072:
2069:
2066:
2063:
2061:
2060:
2057:
2054:
2052:
2049:
2046:
2044:
2043:
2040:
2037:
2035:
2032:
2029:
2026:
2023:
2018:
2015:
2014:
2010:
2008:
2005:
2003:
2002:
1999:
1996:
1993:
1991:
1990:
1987:
1986:charge parity
1984:
1982:
1978:
1976:
1975:
1972:
1969:
1966:
1964:
1961:
1960:
1957:
1954:
1950:
1947:
1943:
1938:
1935:
1930:
1929:Lorentz-boost
1927:
1925:
1924:
1921:
1917:
1913:
1909:
1906:
1903:
1898:
1895:
1893:
1892:
1889:
1885:
1882:
1879:
1874:
1871:
1869:
1868:
1865:
1861:
1858:
1855:
1850:
1847:
1845:
1840:
1839:
1836:
1833:
1831:
1828:
1825:
1824:
1821:
1818:
1816:
1812:
1807:
1803:
1797:
1789:
1787:
1785:
1781:
1777:
1773:
1769:
1765:
1760:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1728:
1723:
1718:
1712:
1707:
1705:
1701:
1700:Lorentz group
1696:
1692:
1688:
1683:
1681:
1677:
1671:
1667:
1661:
1653:
1651:
1646:
1638:
1636:
1634:
1630:
1623:
1622:Supersymmetry
1616:Supersymmetry
1615:
1607:
1597:
1591:
1588:This section
1586:
1577:
1576:
1573:
1571:
1567:
1563:
1559:
1549:
1545:
1541:
1537:
1534:
1533:
1529:
1525:
1522:
1520:
1516:
1513:
1512:
1511:
1509:
1505:
1499:
1498:Wu experiment
1491:
1486:
1482:
1478:
1477:
1473:
1470:
1448:
1442:
1429:
1418:
1417:
1413:
1410:
1394:
1391:
1371:
1348:
1345:
1339:
1336:
1332:
1324:
1308:
1305:
1298:
1290:
1289:
1288:Time reversal
1285:
1284:
1283:
1281:
1277:
1272:
1266:
1258:
1256:
1254:
1250:
1246:
1241:
1239:
1235:
1231:
1230:vector fields
1228:
1220:
1216:
1213:
1210:
1206:
1203:structure of
1202:
1198:
1195:
1192:
1188:
1184:
1180:
1176:
1173:
1170:
1169:
1164:
1160:
1156:
1152:
1148:
1147:
1143:
1133:
1124:
1115:
1110:
1109:
1105:
1101:
1094:
1086:
1081:
1077:
1074:
1069:
1065:
1060:
1056:
1051:
1047:
1043:
1038:
1034:
1033:
1029:
1028:
1027:
1025:
1021:
1017:
1013:
1009:
1005:
993:
988:
986:
981:
979:
974:
973:
971:
970:
965:
962:
960:
957:
956:
955:
954:
946:
943:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
907:
900:
899:
891:
888:
886:
883:
881:
878:
876:
873:
872:
868:
862:
861:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
824:
820:
815:
814:
806:
803:
801:
798:
793:
790:
788:
785:
784:
782:
777:
774:
772:
769:
768:
766:
764:
761:
759:
756:
755:
751:
746:
745:
737:
734:
732:
729:
727:
724:
719:
716:
714:
711:
710:
708:
706:
703:
701:
698:
696:
693:
692:
688:
683:
682:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
640:
636:
631:
630:
619:
613:
611:
605:
603:
597:
595:
589:
587:
581:
580:
579:
578:
573:
568:
566:
560:
558:
556:
550:
548:
546:
540:
538:
536:
530:
529:
528:
527:
522:
517:
512:
511:
502:
498:
495:
492:
488:
485:
482:
478:
475:
472:
468:
465:
462:
458:
455:
452:
448:
445:
442:
438:
435:
434:
430:
425:
424:
420:
416:
415:
412:
408:
404:
400:
392:
390:
388:
384:
376:
374:
372:
368:
364:
360:
356:
352:
348:
340:
338:
333:
326:
320:
318:
309:
302:
298:
278:
271:
264:
260:
256:
251:
249:
245:
241:
237:
232:
230:
222:
220:
218:
214:
209:
207:
203:
195:
193:
191:
187:
183:
180:known as the
179:
175:
170:
168:
163:
161:
160:
155:
154:finite groups
151:
147:
143:
142:
137:
133:
128:
126:
122:
118:
110:
106:
101:
91:
88:
80:
77:February 2018
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
3910:
3888:CPT symmetry
3768:
3745:
3722:
3669:
3665:
3641:
3623:
3599:
3578:
3553:
3523:
3499:
3476:
3455:
3447:Lederman, L.
3415:
3328:
3304:
3299:
3295:
3293:
3124:
3059:
3054:
3028:
2863:
2855:
2853:
2743:
2323:vector field
2223:
2208:
2184:quark flavor
2133:weak isospin
1955:
1952:
1948:
1945:
1941:
1919:
1915:
1911:
1887:
1863:
1819:
1799:
1761:
1749:strong force
1726:
1721:
1719:
1708:
1686:
1684:
1680:Group theory
1673:
1648:
1625:
1601:
1594:Please help
1589:
1562:CP violation
1558:CPT symmetry
1554:
1519:antiparticle
1501:
1474:
1469:mirror image
1414:
1286:
1280:interchanges
1279:
1275:
1270:
1268:
1242:
1224:
1214:
1196:
1174:
1166:
1144:
1131:
1122:
1113:
1106:
1099:
1092:
1090:and also at
1084:
1079:
1075:
1067:
1063:
1058:
1049:
1045:
1041:
1036:
1030:
1023:
1019:
1015:
1003:
1001:
945:Armand Borel
930:Hermann Weyl
731:Loop algebra
713:Killing form
687:Lie algebras
564:
554:
544:
534:
500:
490:
480:
470:
460:
450:
440:
411:Lie algebras
387:continuously
380:
362:
361:, whereas a
354:
350:
346:
344:
331:
324:
321:
316:
307:
300:
296:
269:
262:
258:
254:
252:
247:
243:
233:
228:
226:
210:
205:
199:
171:
164:
157:
139:
131:
129:
116:
114:
83:
74:
55:
3883:CP symmetry
2219:Lie algebra
2205:Mathematics
2169:quark color
2071:hypercharge
2025:coordinates
1998:time parity
1878:homogeneity
1854:homogeneity
1772:unification
1492:C, P, and T
1276:reflections
1159:determinant
1057:parameters
1002:Continuous
925:Ălie Cartan
771:Root system
575:Exceptional
202:temperature
109:FCC lattice
69:introducing
3926:Categories
3865:T-symmetry
3860:P-symmetry
3855:C-symmetry
3765:Wigner, E.
3619:Earman, J.
3574:Earman, J.
3451:Hill, C.T.
3427:2205.09545
3407:References
3311:YangâMills
2215:commutator
2141:P Ă SU(2)
1830:Invariance
1780:weak force
1691:Lie groups
1664:See also:
1643:See also:
1536:T-symmetry
1524:P-symmetry
1515:C-symmetry
1253:isometries
910:Sophus Lie
903:Scientists
776:Weyl group
497:Symplectic
457:Orthogonal
407:Lie groups
377:Continuous
275:the total
229:invariance
186:invariance
150:Lie groups
146:reflection
132:continuous
52:references
3906:Pin group
3901:Chirality
3767:(1970) .
3725:. Wiley.
3692:CiteSeerX
3679:1111.0658
3453:(2011) .
3323:invariant
3274:μ
3264:ν
3254:ν
3243:−
3218:μ
3205:μ
3189:μ
3176:ν
3166:ν
3163:μ
3141:μ
3101:ϕ
3089:Ω
3074:ϕ
3071:δ
3055:different
3051:Lie group
3037:τ
3002:λ
2997:μ
2993:∂
2975:μ
2967:δ
2935:β
2931:ψ
2925:β
2922:α
2918:τ
2902:λ
2885:α
2881:ψ
2877:δ
2837:μ
2824:ν
2814:ν
2811:μ
2789:μ
2718:ν
2708:μ
2704:∂
2689:ν
2667:μ
2657:ν
2653:∂
2638:ν
2616:μ
2608:δ
2576:β
2572:ψ
2566:β
2563:α
2558:ν
2555:μ
2551:σ
2536:ν
2526:μ
2522:∂
2504:α
2500:ψ
2494:μ
2490:∂
2475:μ
2453:α
2449:ψ
2445:δ
2415:ϕ
2410:μ
2406:∂
2391:μ
2371:ϕ
2368:δ
2300:ψ
2268:ϕ
2022:spacetime
1604:June 2015
1452:→
1443:−
1440:→
1433:→
1409:parabolic
1392:−
1349:¨
1306:−
1303:→
1205:spacetime
1163:inversion
787:Real form
673:Euclidean
524:Classical
393:Spacetime
367:spacetime
359:spacetime
206:invariant
178:spacetime
134:(such as
3952:Symmetry
3803:Symmetry
3743:(1989).
3714:14475702
3621:(2002),
3576:(eds.).
3521:(2007).
3498:(2000).
3334:See also
2145:G-parity
1934:isotropy
1902:isotropy
1778:and the
1689:(called
1633:fermions
1548:Big Bang
1481:crystals
1259:Discrete
1201:geodesic
1053:for any
959:Glossary
653:Poincaré
141:discrete
136:rotation
117:symmetry
3684:Bibcode
3519:Zee, A.
3057:types.
2115:isospin
1544:entropy
867:physics
648:Lorentz
477:Unitary
319:-axis.
294:
282:
144:(e.g.,
65:improve
3775:
3753:
3729:
3712:
3694:
3649:
3607:
3586:
3561:
3531:
3506:
3484:
3463:
3345:Charge
3343:&
3309:super-
3029:where
2860:matrix
2854:where
2291:spinor
2259:scalar
2213:. The
2211:fields
2099:U(2)
1860:energy
1826:Class
1729:theory
1629:bosons
1551:versa.
1249:metric
1227:smooth
643:Circle
347:global
103:First
54:, but
3710:S2CID
3674:arXiv
3628:(PDF)
3422:arXiv
3294:with
2178:SU(3)
2124:SU(2)
1727:gauge
1722:local
1676:group
1232:on a
1141:room.
1008:space
718:Index
351:local
156:(see
119:of a
3773:ISBN
3751:ISBN
3727:ISBN
3647:ISBN
3605:ISBN
3584:ISBN
3559:ISBN
3529:ISBN
3504:ISBN
3482:ISBN
3461:ISBN
2080:U(1)
2065:U(1)
2048:U(1)
2031:U(1)
1668:and
1631:and
1502:The
1153:and
1061:and
1055:real
1012:time
1010:and
668:Loop
409:and
353:. A
330:and
217:axis
115:The
3801:: "
3702:doi
3307:= 4
2357:):
2321:or
2007:CPT
1979:C,
1786:).
1782:in
1774:of
1759:.)
1706:).
1506:of
1278:or
1076:mgh
499:Sp(
489:SU(
469:SO(
449:SL(
439:GL(
349:or
162:).
107:of
3928::
3878:CP
3708:.
3700:.
3690:.
3682:.
3670:38
3668:.
3664:.
3449:;
2289:,
2221:.
2027:)
1951:â
1944:=
1936:)
1918:Ă
1914:=
1904:)
1880:)
1856:)
1817:.
1717:.
1678:.
1572:.
1560:.
1364:,
1282:.
1269:A
1255:.
1130:+
1121:â
1098:+
1066:+
1048:+
1044:â
479:U(
459:O(
373:.
306:+
192:.
127:.
3838:e
3831:t
3824:v
3781:.
3759:.
3735:.
3716:.
3704::
3686::
3676::
3655:.
3635:.
3613:.
3592:.
3567:.
3537:.
3512:.
3490:.
3469:.
3430:.
3424::
3305:N
3300:K
3296:D
3279:,
3270:x
3260:x
3250:K
3246:2
3238:2
3233:|
3228:x
3224:|
3214:K
3210:+
3201:x
3197:D
3194:+
3185:P
3181:+
3172:x
3159:M
3155:=
3152:)
3149:x
3146:(
3137:h
3110:)
3107:x
3104:(
3098:)
3095:x
3092:(
3086:=
3083:)
3080:x
3077:(
3014:,
3011:)
3008:x
3005:(
2989:=
2986:)
2983:x
2980:(
2971:A
2946:)
2943:x
2940:(
2914:.
2911:)
2908:x
2905:(
2899:=
2896:)
2893:x
2890:(
2864:P
2856:M
2833:P
2829:+
2820:x
2807:M
2803:=
2800:)
2797:x
2794:(
2785:h
2761:)
2758:x
2755:(
2752:h
2729:)
2726:x
2723:(
2714:h
2700:)
2697:x
2694:(
2685:A
2681:+
2678:)
2675:x
2672:(
2663:A
2649:)
2646:x
2643:(
2634:h
2630:=
2627:)
2624:x
2621:(
2612:A
2587:)
2584:x
2581:(
2547:)
2544:x
2541:(
2532:h
2518:+
2515:)
2512:x
2509:(
2486:)
2483:x
2480:(
2471:h
2467:=
2464:)
2461:x
2458:(
2424:)
2421:x
2418:(
2402:)
2399:x
2396:(
2387:h
2383:=
2380:)
2377:x
2374:(
2341:)
2338:x
2335:(
2332:A
2309:)
2306:x
2303:(
2277:)
2274:x
2271:(
2241:)
2238:x
2235:(
2232:h
2127:L
2083:Y
1956:r
1953:E
1949:p
1946:t
1942:N
1932:(
1920:p
1916:r
1912:L
1900:(
1888:p
1876:(
1864:E
1852:(
1715:3
1713:S
1606:)
1602:(
1598:.
1592:.
1530:.
1487:.
1471:.
1449:r
1430:r
1395:t
1372:t
1346:r
1340:m
1337:=
1333:F
1309:t
1299:t
1211:.
1193:.
1171:.
1136:â
1132:a
1127:â
1123:r
1118:â
1114:r
1100:a
1096:0
1093:t
1088:0
1085:t
1080:h
1068:a
1064:t
1059:t
1050:a
1046:t
1042:t
1037:t
991:e
984:t
977:v
617:8
615:E
609:7
607:E
601:6
599:E
593:4
591:F
585:2
583:G
565:n
562:D
555:n
552:C
545:n
542:B
535:n
532:A
503:)
501:n
493:)
491:n
483:)
481:n
473:)
471:n
463:)
461:n
453:)
451:n
443:)
441:n
335:2
332:v
328:1
325:v
317:y
313:)
311:2
308:v
304:1
301:v
299:(
297:m
291:2
288:/
285:1
273:2
270:v
266:1
263:v
259:x
255:m
248:r
244:r
90:)
84:(
79:)
75:(
61:.
34:.
20:)
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