Symmetry breaking
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In
In an infinite system (Minkowski spacetime) symmetry breaking occurs, however in a finite system (that is, any real super-condensed system), the system is less predictable, but in many cases quantum tunneling occurs.[2][3] Symmetry breaking and tunneling relate through the collapse of a particle into non-symmetric state as it seeks a lower energy.[4]
Symmetry breaking can be distinguished into two types,
Non-technical description
This section describes spontaneous symmetry breaking. This is the idea that for a physical system, the lowest energy configuration (the vacuum state) is not the most symmetric configuration of the system. Roughly speaking there are three types of symmetry that can be broken: discrete, continuous and gauge, ordered in increasing technicality.
An example of a system with discrete symmetry is given by the figure with the red graph: consider a particle moving on this graph, subject to gravity. A similar graph could be given by the function . This system is symmetric under reflection in the y-axis. There are three possible stationary states for the particle: the top of the hill at , or the bottom, at . When the particle is at the top, the configuration respects the reflection symmetry: the particle stays in the same place when reflected. However, the lowest energy configurations are those at . When the particle is in either of these configurations, it is no longer fixed under reflection in the y-axis: reflection swaps the two vacuum states.
An example with continuous symmetry is given by a 3d analogue of the previous example, from rotating the graph around an axis through the top of the hill, or equivalently given by the graph . This is essentially the graph of the
Gauge symmetry breaking is the most subtle, but has important physical consequences. Roughly speaking, for the purposes of this section a gauge symmetry is an assignment of systems with continuous symmetry to every point in
Spontaneous symmetry breaking
In spontaneous symmetry breaking (SSB), the equations of motion of the system are invariant, but any vacuum state (lowest energy state) is not.
For an example with two-fold symmetry, if there is some atom which has two vacuum states, occupying either one of these states breaks the two-fold symmetry. This act of selecting one of the states as the system reaches a lower energy is SSB. When this happens, the atom is no longer symmetric (reflectively symmetric) and has collapsed into a lower energy state.
Such a symmetry breaking is parametrized by an
In the Lagrangian setting of Quantum field theory (QFT), the Lagrangian is a functional of quantum fields which is invariant under the action of a symmetry group . However, the vacuum expectation value formed when the particle collapses to a lower energy may not be invariant under . In this instance, it will partially break the symmetry of , into a subgroup . This is spontaneous symmetry breaking.
Within the context of gauge symmetry however, SSB is the phenomenon by which
Further, in this context the usage of 'symmetry breaking' while standard, is a misnomer, as gauge 'symmetry' is not really a symmetry but a redundancy in the description of the system. Mathematically, this redundancy is a choice of
Spontaneous symmetry breaking is also associated with
Explicit symmetry breaking
In
In the Hamiltonian setting, this is often studied when the Hamiltonian can be written .
Here is a 'base Hamiltonian', which has some manifest symmetry. More explicitly, it is symmetric under the action of a (Lie) group . Often this is an integrable Hamiltonian.
The is a perturbation or interaction Hamiltonian. This is not invariant under the action of . It is often proportional to a small, perturbative parameter.
This is essentially the paradigm for perturbation theory in quantum mechanics. An example of its use is in finding the fine structure of atomic spectra.
Examples
Symmetry breaking can cover any of the following scenarios:
- The breaking of an exact symmetry of the underlying laws of physics by the apparently random formation of some structure;
- A situation in physics in which a minimal energy state has less symmetry than the system itself;
- Situations where the actual state of the system does not reflect the underlying symmetries of the dynamics because the manifestly symmetric state is unstable (stability is gained at the cost of local asymmetry);
- Situations where the equations of a theory may have certain symmetries, though their solutions may not (the symmetries are "hidden").
One of the first cases of broken symmetry discussed in the physics literature is related to the form taken by a uniformly rotating body of
See also
References
- S2CID 4568832– via SpringerLink.
- ^ PMID 11607718.
- .
- ^ Castellani, Elena; Teh, Nicholas; Brading, Katherine (2017-12-14). Edward, Zalta (ed.). "Symmetry and symmetry breaking". Stanford Encyclopedia of Philosophy (Fall 2021 ed.). Metaphysics Research Lab, Stanford University.
- ISBN 9780199233991. Retrieved 2023-03-01.
- .
- ^ Liouville, J. (1834). "Sur la figure d'une masse fluide homogène, en équilibre et douée d'un mouvement de rotation". Journal de l'École Polytechnique (14): 289–296.
External links
- Quotations related to Symmetry breaking at Wikiquote