In QED a vertex always has three lines attached to it: one bosonic line, one fermionic line with arrow toward the vertex, and one fermionic line with arrow away from the vertex.

For the QED interaction Lagrangian,, describing the interaction of a fermionic field with a bosonic gauge field, the Feynman rules can be formulated in coordinate space as follows:

In the case of the open bosonic string a gauge-unfixed action with the appropriate symmetries and equations of motion was originally obtained by André Neveu, Hermann Nicolai and Peter C. West.

The formulation amalgams bosonic closed string field theory with Berkovits ' superstring field theory.

The polariton is a bosonic quasiparticle, and should not be confused with the polaron, a fermionic one, e. g. an electron plus attached phonon cloud.

* In one-dimensional many-body quantum systems with fermionic degrees of freedom, low energy excited states can be approximated as coherent states of a bosonic field operator that creates particle-hole excitations.

It is demonstrated that this proposal can be realized in present experiments with bosonic mixtures in an optical lattice that features simple on-site interactions.

This implies that the closed bosonic string compactified on a radius is equivalent to the theory with radius.

) Gauge theories were not invented starting with bosonic or fermionic strings ; they originated from a different area of physics, and have become quite useful in their own right.

Nevertheless, bosonic string theory remains a very useful " toy model " to understand many general features of perturbative string theory, and string theory textbooks usually start with the bosonic string.

In string theory, the left-moving and the right-moving excitations almost do not interact with each other, and it is possible to construct a string theory whose left-moving ( counter-clockwise ) excitations " think " that they live on a bosonic string propagating in D = 26 dimensions, while the right-moving ( clock-wise ) excitations " think " that they belong to a superstring in D = 10 dimensions.

In general, anti-commuting ghosts are associated with fermionic symmetries, while commuting ghosts are associated with bosonic symmetries.

( Closed fermion loops have an extra − 1 associated with them ; bosonic loops don't.

The polaron, a fermionic quasiparticle, should not be confused with the polariton, a bosonic quasiparticle, corresponding, e. g., to a hybridized state between a photon and an optical phonon.

In physics, a supermanifold is a manifold with both bosonic and fermionic coordinates.

As it turns out, string theories involve higher-dimensional spaces than the 3D world with which we are familiar ; bosonic string theory requires 25 spatial dimensions and one time axis.

The Fourier transform is essentially bosonic ; it works because it is consistent with the superposition principle and related interference patterns.

i. e. they are bosonic with respect to themselves but fermionic with respect to each other.

This does not mean that chemistry is never involved with plasma or nuclear sciences or even bosonic fields nowadays, since areas such as Quantum Chemistry and Nuclear Chemistry are currently well developed and formally recognized sub-fields of study under the Chemical sciences ( Chemistry ), but what is now formally recognized as subject of study under the Chemistry category as a science is always based on the use of concepts that describe or explain phenomena either from matter or to matter in the atomic or molecular scale, including the study of the behavior of many molecules as an aggregate or the study of the effects of a single proton on a single atom, but excluding phenomena that deal with different ( more " exotic ") types of matter ( e. g. Bose-Einstein condensate, Higgs Boson, dark matter, naked singularity, etc.

The smallest superspace is a point which contains neither bosonic nor fermionic directions.

In the nonsupersymmetric version the action is invariant under a similar Z < sub > 2 </ sub > symmetry because the matter fields are all fermionic and thus must appear in the action in pairs, while the Higgs fields are bosonic.

then only the symmetric part of contributes, so that and the field will create bosonic particles.

Below a certain temperature, most of the particles in a bosonic system will occupy the ground state ( the state of lowest energy ).

We will use the following conventions ; the spatial ( both bosonic and fermionic ) indices will be indicated by M, N, ....

The bosonic spatial indices will be indicated by μ, ν, ..., the left-handed Weyl spatial indices by α, β ,..., and the right-handed Weyl spatial indices by,, ....

We will now examine the normal ordering of bosonic creation and annihilation operator products.

These have some bosonic properties, while bosons, at sufficiently low temperature, can form a large Bose-Einstein condensate.

Einstein demonstrated that cooling bosonic atoms to a very low temperature would cause them to fall ( or " condense ") into the lowest accessible quantum state, resulting in a new form of matter.

For example, in the bosonic closed string case, the worldsheet scattering diagrams naturally take a Feynman diagram-like form, being built from two ingredients, a propagator,

For example, in the case of the bosonic open string theory in 26-dimensional flat spacetime, a general element of the Fock-space of the BRST quantized string takes the form ( in radial quantization in the upper half plane ),

The first is very similar in form to its bosonic cousin and is known as modified cubic superstring field theory.

The action takes a very similar form to bosonic action,

A helium-3 atom is a fermion and at very low temperatures, they form two-atom Cooper pairs which are bosonic and condense into a superfluid.

This equation, which is in the same form as the bosonic case above, is used in defining the contractions used in Wick's theorem.

The vertices might be connected by a bosonic or fermionic propagator.

In nonrelativistic quantum mechanics all particles are either bosons or fermions ; in relativistic quantum theories also " supersymmetric " theories exist, where a particle is a linear combination of a bosonic and a fermionic part.

This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free.

According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute.

Supersymmetry reduces the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions.

Here is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic or fermionic statistics.

Because supersymmetry guarantees the cancellation of fermionic vs. bosonic non-zero modes in the instanton background, the involved't Hooft computation of the instanton saddle point reduces to an integration over zero modes.

Defining as a general annihilation ( creation ) operator that could be either fermionic or bosonic, the real space representation of the operators defines the quantum field operators and by

The ordinary space dimensions correspond to bosonic degrees of freedom, the anticommuting dimensions to fermionic degrees of freedom.

Other trivial examples include the n-dimensional real plane R < sup > n </ sup >, which is a vector space extending in n real, bosonic directions and no fermionic directions.

The bosonic part of the supersymmetry algebra is the Poincaré algebra, while the fermionic part is constructed using spinors of Grassmann numbers.

In 4D N = 1 SUGRA, we have a 4 | 4 real differentiable supermanifold M, i. e. we have 4 real bosonic dimensions and 4 real fermionic dimensions.

As in the nonsupersymmetric case, we have a Spin ( 3, 1 ) principal bundle over M. We have an R < sup > 4 | 4 </ sup > vector bundle T over M. The fiber of T transforms under the local Lorentz group as follows ; the four real bosonic dimensions transform as a vector and the four real fermionic dimensions transform as a Majorana spinor.

It is far more difficult to produce a fermionic superfluid than a bosonic one, because the Pauli exclusion principle prohibits fermions from occupying the same quantum state.