Pauli matrices

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Wolfgang Pauli (1900–1958), c. 1924. Pauli received the Nobel Prize in physics in 1945, nominated by Albert Einstein, for the Pauli exclusion principle.

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.

$${\displaystyle \begin{array}{rl}{\sigma }_1={\sigma }_{\mathrm{x}}& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\\ {\sigma }_2={\sigma }_{\mathrm{y}}& =\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\\ {\sigma }_3={\sigma }_{\mathrm{z}}& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\end{array}}$$

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal / vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ₀ ), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices. This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex 2 dimensional Hilbert space. In the context of Pauli's work, σ_k represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space ${\displaystyle {\mathbb{R}}^3.}$

The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ₁, iσ₂, iσ₃ form a basis for the real Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$, which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices σ₁, σ₂, σ₃ is isomorphic to the Clifford algebra of ${\displaystyle {\mathbb{R}}^3,}$ and the (unital associative) algebra generated by iσ₁, iσ₂, iσ₃ functions identically (is isomorphic) to that of quaternions (${\displaystyle \mathbb{H}}$).

Algebraic properties

Cayley table; the entry shows the value of the row times the column.

×	${\displaystyle {\sigma }_{\mathrm{x}}}$	${\displaystyle {\sigma }_{\mathrm{y}}}$	${\displaystyle {\sigma }_{\mathrm{z}}}$
${\displaystyle {\sigma }_{\mathrm{x}}}$	${\displaystyle I}$	${\displaystyle i{\sigma }_{\mathrm{z}}}$	${\displaystyle -i{\sigma }_{\mathrm{y}}}$
${\displaystyle {\sigma }_{\mathrm{y}}}$	${\displaystyle -i{\sigma }_{\mathrm{z}}}$	${\displaystyle I}$	${\displaystyle i{\sigma }_{\mathrm{x}}}$
${\displaystyle {\sigma }_{\mathrm{z}}}$	${\displaystyle i{\sigma }_{\mathrm{y}}}$	${\displaystyle -i{\sigma }_{\mathrm{x}}}$	${\displaystyle I}$

All three of the Pauli matrices can be compacted into a single expression:

${\displaystyle {\sigma }_j=\left(\begin{array}{cc}{\delta }_{j3}& {\delta }_{j1}-i{\delta }_{j2}\\ {\delta }_{j1}+i{\delta }_{j2}& -{\delta }_{j3}\end{array}\right)}$

where the solution to i² = −1 is the "imaginary unit", and δ_jk is the Kronecker delta, which equals +1 if j = k and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of j = 1, 2, 3 , in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are involutory:

${\displaystyle {\sigma }_1^2={\sigma }_2^2={\sigma }_3^2=-i{\sigma }_1{\sigma }_2{\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=I}$

where I is the identity matrix.

The determinants and traces of the Pauli matrices are:

${\displaystyle \begin{array}{rl}det{\sigma }_j& =-1,\\ \mathrm{tr}{\sigma }_j& =0.\end{array}}$

From which, we can deduce that each matrix σ_j has eigenvalues +1 and −1.

With the inclusion of the identity matrix, I (sometimes denoted σ₀ ), the Pauli matrices form an orthogonal basis (in the sense of Hilbert–Schmidt) of the Hilbert space of 2 × 2 Hermitian matrices, ${\displaystyle {\mathcal{H}}_2}$, over ${\displaystyle \mathbb{R}}$, and the Hilbert space of all complex 2 × 2 matrices, ${\displaystyle {\mathcal{M}}_{2,2}(\mathbb{C})}$, over ${\displaystyle \mathbb{C}}$.

Commutation and anti-commutation relations

The Pauli matrices obey the following commutation relations:

${\displaystyle [{\sigma }_i,{\sigma }_j]=2i{\epsilon }_{ijk}{\sigma }_k,}$

where the structure constant ε_ijk is the Levi-Civita symbol and Einstein summation notation is used.

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra ${\displaystyle ({\mathbb{R}}^3,\times )\cong \mathfrak{s}\mathfrak{u}(2)\cong \mathfrak{s}\mathfrak{o}(3).}$

They also satisfy the anticommutation relations:

${\displaystyle \{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}I,}$

where ${\displaystyle \{{\sigma }_i,{\sigma }_j\}}$ is defined as ${\displaystyle {\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i,}$ and δ_ij is the Kronecker delta. I denotes the 2 × 2 identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for ${\displaystyle {\mathbb{R}}^3,}$ denoted ${\displaystyle {\mathrm{C}\mathrm{l}}_3(\mathbb{R}).}$

The usual construction of generators ${\sigma }_{ij}=\frac{1}{4}\left[{\sigma }_i,{\sigma }_j\right],$ of ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.

A few explicit commutators and anti-commutators are given below as examples:

Commutators	Anticommutators
${\displaystyle \begin{array}{rl}\left[{\sigma }_1,{\sigma }_1\right]& =0\\ \left[{\sigma }_1,{\sigma }_2\right]& =2i{\sigma }_3\\ \left[{\sigma }_2,{\sigma }_3\right]& =2i{\sigma }_1\\ \left[{\sigma }_3,{\sigma }_1\right]& =2i{\sigma }_2\end{array}}$	${\displaystyle \begin{array}{rl}\left\{{\sigma }_1,{\sigma }_1\right\}& =2I\\ \left\{{\sigma }_1,{\sigma }_2\right\}& =0\\ \left\{{\sigma }_1,{\sigma }_3\right\}& =0\\ \left\{{\sigma }_3,{\sigma }_1\right\}& =0\end{array}}$

Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are:

${\displaystyle \begin{array}{rlrl}{\psi }_{x+}& =\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right],& {\psi }_{x-}& =\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -1\end{array}\right],\\ {\psi }_{y+}& =\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ i\end{array}\right],& {\psi }_{y-}& =\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ -i\end{array}\right],\\ {\psi }_{z+}& =\left[\begin{array}{c}1\\ 0\end{array}\right],& {\psi }_{z-}& =\left[\begin{array}{c}0\\ 1\end{array}\right].\end{array}}$

Pauli vectors

The Pauli vector is defined by

$${\displaystyle \overrightarrow{\sigma }={\sigma }_1{\hat{x}}_1+{\sigma }_2{\hat{x}}_2+{\sigma }_3{\hat{x}}_3,}$$

where ${\displaystyle {\hat{x}}_1}$, ${\displaystyle {\hat{x}}_2}$, and ${\displaystyle {\hat{x}}_3}$ are an equivalent notation for the more familiar ${\displaystyle \hat{x}}$, ${\displaystyle \hat{y}}$, and ${\displaystyle \hat{z}}$.

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows,

$${\displaystyle \begin{array}{rl}\overrightarrow{a}\cdot \overrightarrow{\sigma }& =\left(a_k{\hat{x}}_k\right)\cdot \left({\sigma }_{\ell }{\hat{x}}_{\ell }\right)=a_k{\sigma }_{\ell }{\hat{x}}_k\cdot {\hat{x}}_{\ell }\\ \\ & =a_k{\sigma }_{\ell }{\delta }_{k\ell }=a_k{\sigma }_k\\ \\ & =a_1\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)+a_2\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)+a_3\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)=\left(\begin{array}{cc}{a}_3& a_1-i{a}_2\\ a_1+i{a}_2& -a_3\end{array}\right)\end{array}}$$

using Einstein's summation convention.

More formally, this defines a map from ${\displaystyle {\mathbb{R}}^3}$ to the vector space of traceless Hermitian ${\displaystyle 2\times 2}$ matrices. This map encodes structures of ${\displaystyle {\mathbb{R}}^3}$ as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory.

Another way to view the Pauli vector is as a ${\displaystyle 2\times 2}$ Hermitian traceless matrix-valued dual vector, that is, an element of ${\displaystyle {\text{Mat}}_{2\times 2}(\mathbb{C})\otimes ({\mathbb{R}}^3{)}^{\ast }}$ which maps ${\displaystyle \overrightarrow{a}\mapsto \overrightarrow{a}\cdot \overrightarrow{\sigma }.}$

Completeness relation

Each component of ${\displaystyle \overrightarrow{a}}$ can be recovered from the matrix (see completeness relation below)

$${\displaystyle \frac{1}{2}\mathrm{tr}((\overrightarrow{a}\cdot \overrightarrow{\sigma })\overrightarrow{\sigma })=\overrightarrow{a}.}$$

This constitutes an inverse to the map ${\displaystyle \overrightarrow{a}\mapsto \overrightarrow{a}\cdot \overrightarrow{\sigma }}$, making it manifest that the map is a bijection.

Determinant

The norm is given by the determinant (up to a minus sign)

$${\displaystyle det(\overrightarrow{a}\cdot \overrightarrow{\sigma })=-\overrightarrow{a}\cdot \overrightarrow{a}=-{\left|\overrightarrow{a}\right|}^2.}$$

Then considering the conjugation action of an ${\displaystyle \text{SU}(2)}$ matrix ${\displaystyle U}$ on this space of matrices,

${\displaystyle U\ast \overrightarrow{a}\cdot \overrightarrow{\sigma }:=U\overrightarrow{a}\cdot \overrightarrow{\sigma }{U}^{-1},}$

we find ${\displaystyle det(U\ast \overrightarrow{a}\cdot \overrightarrow{\sigma })=det(\overrightarrow{a}\cdot \overrightarrow{\sigma }),}$ and that ${\displaystyle U\ast \overrightarrow{a}\cdot \overrightarrow{\sigma }}$ is Hermitian and traceless. It then makes sense to define ${\displaystyle U\ast \overrightarrow{a}\cdot \overrightarrow{\sigma }={\overrightarrow{a}}^{\prime }\cdot \overrightarrow{\sigma }}$ where ${\displaystyle {\overrightarrow{a}}^{\prime }}$ has the same norm as ${\displaystyle \overrightarrow{a},}$ and therefore interpret ${\displaystyle U}$ as a rotation of 3-dimensional space. In fact, it turns out that the special restriction on ${\displaystyle U}$ implies that the rotation is orientation preserving. This allows the definition of a map ${\displaystyle R:\mathrm{S}\mathrm{U}(2)\to \mathrm{S}\mathrm{O}(3)}$ given by

${\displaystyle U\ast \overrightarrow{a}\cdot \overrightarrow{\sigma }={\overrightarrow{a}}^{\prime }\cdot \overrightarrow{\sigma }=:(R(U)\overrightarrow{a})\cdot \overrightarrow{\sigma },}$

where ${\displaystyle R(U)\in \mathrm{S}\mathrm{O}(3).}$ This map is the concrete realization of the double cover of ${\displaystyle \mathrm{S}\mathrm{O}(3)}$ by ${\displaystyle \mathrm{S}\mathrm{U}(2),}$ and therefore shows that ${\displaystyle \text{SU}(2)\cong \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(3).}$ The components of ${\displaystyle R(U)}$ can be recovered using the tracing process above:

${\displaystyle R(U{)}_{ij}=\frac{1}{2}\mathrm{tr}\left({\sigma }_{i}U{\sigma }_j{U}^{-1}\right)}$

Cross-product

The cross-product is given by the matrix commutator (up to a factor of ${\displaystyle 2i}$)

$${\displaystyle \left[\overrightarrow{a}\cdot \overrightarrow{\sigma },\overrightarrow{b}\cdot \overrightarrow{\sigma }\right]=2i(\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{\sigma }.}$$

In fact, the existence of a norm follows from the fact that ${\displaystyle {\mathbb{R}}^3}$ is a Lie algebra: see Killing form.

This cross-product can be used to prove the orientation-preserving property of the map above.

Eigenvalues and eigenvectors

The eigenvalues of ${\displaystyle \overrightarrow{a}\cdot \overrightarrow{\sigma }}$ are ${\displaystyle \pm |\overrightarrow{a}|.}$ This follows immediately from tracelessness and explicitly computing the determinant.

More abstractly, without computing the determinant which requires explicit properties of the Pauli matrices, this follows from ${\displaystyle (\overrightarrow{a}\cdot \overrightarrow{\sigma }{)}^2-|\overrightarrow{a}{|}^2=0,}$ since this can be factorised into ${\displaystyle (\overrightarrow{a}\cdot \overrightarrow{\sigma }-|\overrightarrow{a}|)(\overrightarrow{a}\cdot \overrightarrow{\sigma }+|\overrightarrow{a}|)=0.}$ A standard result in linear algebra (a linear map which satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies ${\displaystyle \overrightarrow{a}\cdot \overrightarrow{\sigma }}$ is diagonal with possible eigenvalues ${\displaystyle \pm |\overrightarrow{a}|.}$ The tracelessness of ${\displaystyle \overrightarrow{a}\cdot \overrightarrow{\sigma }}$ means it has exactly one of each eigenvalue.

Its normalized eigenvectors are

$${\displaystyle {\psi }_{+}=\frac{1}{\sqrt{2\left|\overrightarrow{a}\right|(a_3+\left|\overrightarrow{a}\right|)}}\left[\begin{array}{c}{a}_3+\left|\overrightarrow{a}\right|\\ a_1+i{a}_2\end{array}\right];{\psi }_{-}=\frac{1}{\sqrt{2|\overrightarrow{a}|(a_3+|\overrightarrow{a}|)}}\left[\begin{array}{c}i{a}_2-a_1\\ a_3+|\overrightarrow{a}|\end{array}\right].}$$

These expressions become singular for ${\displaystyle a_3\to -1}$. They can be rescued by letting ${\displaystyle \overrightarrow{a}=(ϵ,0,-(1-{ϵ}^2/2))}$ and taking the limit ${\displaystyle ϵ\to 0}$, which yields the correct eigenvectors (0,1) and (1,0) of ${\displaystyle {\sigma }_z}$.

Aternatively, one may use spherical coordinates ${\displaystyle \overrightarrow{a}=a(\cos \vartheta \cos \phi ,\cos \vartheta \sin \phi ,\sin \vartheta )}$ to obtain the eigenvectors ${\displaystyle {\psi }_{+}=(\cos (\vartheta /2),\sin (\vartheta /2)\exp (i\phi ))}$ and ${\displaystyle {\psi }_{-}=(-\sin (\vartheta /2)\exp (-i\phi ),\cos (\vartheta /2))}$.

Pauli 4-vector

The Pauli 4-vector, used in spinor theory, is written ${\displaystyle {\sigma }^{\mu }}$ with components

${\displaystyle {\sigma }^{\mu }=(I,\overrightarrow{\sigma }).}$

This defines a map from ${\displaystyle {\mathbb{R}}^{1,3}}$ to the vector space of Hermitian matrices,

${\displaystyle x_{\mu }\mapsto x_{\mu }{\sigma }^{\mu },}$

which also encodes the Minkowski metric (with mostly minus convention) in its determinant:

${\displaystyle det(x_{\mu }{\sigma }^{\mu })=\eta (x,x).}$

This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector

${\displaystyle {\overline{\sigma }}^{\mu }=(I,-\overrightarrow{\sigma }).}$

and allow raising and lowering using the Minkowski metric tensor. The relation can then be written

$${\displaystyle x_{\nu }=\frac{1}{2}\mathrm{tr}({\overline{\sigma }}_{\nu }(x_{\mu }{\sigma }^{\mu })).}$$

Similarly to the Pauli 3-vector case, we can find a matrix group which acts as isometries on ${\displaystyle {\mathbb{R}}^{1,3};}$ in this case the matrix group is ${\displaystyle \mathrm{S}\mathrm{L}(2,\mathbb{C}),}$ and this shows ${\displaystyle \mathrm{S}\mathrm{L}(2,\mathbb{C})\cong \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(1,3).}$ Similarly to above, this can be explicitly realized for ${\displaystyle S\in \mathrm{S}\mathrm{L}(2,\mathbb{C})}$ with components

${\displaystyle \mathrm{\Lambda }(S{)}^{\mu }{}_{\nu }=\frac{1}{2}\mathrm{tr}\left({\overline{\sigma }}_{\nu }S{\sigma }^{\mu }{S}^{†}\right).}$

In fact, the determinant property follows abstractly from trace properties of the ${\displaystyle {\sigma }^{\mu }.}$ For ${\displaystyle 2\times 2}$ matrices, the following identity holds:

${\displaystyle det(A+B)=det(A)+det(B)+\mathrm{tr}(A)\mathrm{tr}(B)-\mathrm{tr}(AB).}$

That is, the 'cross-terms' can be written as traces. When ${\displaystyle A,B}$ are chosen to be different ${\displaystyle {\sigma }^{\mu },}$ the cross-terms vanish. It then follows, now showing summation explicitly, $det\left(\sum _{\mu }{x}_{\mu }{\sigma }^{\mu }\right)=\sum _{\mu }det\left(x_{\mu }{\sigma }^{\mu }\right).$ Since the matrices are ${\displaystyle 2\times 2,}$ this is equal to $\sum _{\mu }{x}_{\mu }^{2}det({\sigma }^{\mu })=\eta (x,x).$

Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives

${\displaystyle \begin{array}{rl}\left[{\sigma }_j,{\sigma }_k\right]+\{{\sigma }_j,{\sigma }_k\}& =({\sigma }_j{\sigma }_k-{\sigma }_k{\sigma }_j)+({\sigma }_j{\sigma }_k+{\sigma }_k{\sigma }_j)\\ 2i{\epsilon }_{jk\ell }{\sigma }_{\ell }+2{\delta }_{jk}I& =2{\sigma }_j{\sigma }_k\end{array}}$

so that,

${\displaystyle {\sigma }_j{\sigma }_k={\delta }_{jk}I+i{\epsilon }_{jk\ell }{\sigma }_{\ell }.}$

Contracting each side of the equation with components of two 3-vectors a_p and b_q (which commute with the Pauli matrices, i.e., a_pσ_q = σ_qa_p) for each matrix σ_q and vector component a_p (and likewise with b_q) yields

${\displaystyle \begin{array}{rl}{a}_j{b}_k{\sigma }_j{\sigma }_k& =a_j{b}_k\left(i{\epsilon }_{jk\ell }{\sigma }_{\ell }+{\delta }_{jk}I\right)\\ a_j{\sigma }_j{b}_k{\sigma }_k& =i{\epsilon }_{jk\ell }{a}_j{b}_k{\sigma }_{\ell }+a_j{b}_k{\delta }_{jk}I\end{array}.}$

Finally, translating the index notation for the dot product and cross product results in

${\displaystyle (\overrightarrow{a}\cdot \overrightarrow{\sigma })(\overrightarrow{b}\cdot \overrightarrow{\sigma })=(\overrightarrow{a}\cdot \overrightarrow{b})I+i(\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{\sigma }}$

(1)

If i is identified with the pseudoscalar σ_xσ_yσ_z then the right hand side becomes ${\displaystyle a\cdot b+a\wedge b}$ which is also the definition for the product of two vectors in geometric algebra.

If we define the spin operator as J = ħ/2σ, then J satisfies the commutation relation:

$${\displaystyle \mathbf{J}\times \mathbf{J}=i\hslash \mathbf{J}}$$

Or equivalently, the Pauli vector satisfies:

$${\displaystyle \frac{\overrightarrow{\sigma }}{2}\times \frac{\overrightarrow{\sigma }}{2}=i\frac{\overrightarrow{\sigma }}{2}}$$

Some trace relations

The following traces can be derived using the commutation and anticommutation relations.

${\displaystyle \begin{array}{rl}\mathrm{tr}\left({\sigma }_j\right)& =0\\ \mathrm{tr}\left({\sigma }_j{\sigma }_k\right)& =2{\delta }_{jk}\\ \mathrm{tr}\left({\sigma }_j{\sigma }_k{\sigma }_{\ell }\right)& =2i{\epsilon }_{jk\ell }\\ \mathrm{tr}\left({\sigma }_j{\sigma }_k{\sigma }_{\ell }{\sigma }_m\right)& =2\left({\delta }_{jk}{\delta }_{\ell m}-{\delta }_{j\ell }{\delta }_{km}+{\delta }_{jm}{\delta }_{k\ell }\right)\end{array}}$

If the matrix σ₀ = I is also considered, these relationships become

$${\displaystyle \begin{array}{rl}\mathrm{tr}\left({\sigma }_{\alpha }\right)& =2{\delta }_{0\alpha }\\ \mathrm{tr}\left({\sigma }_{\alpha }{\sigma }_{\beta }\right)& =2{\delta }_{\alpha \beta }\\ \mathrm{tr}\left({\sigma }_{\alpha }{\sigma }_{\beta }{\sigma }_{\gamma }\right)& =2\sum _{(\alpha \beta \gamma )}{\delta }_{\alpha \beta }{\delta }_{0\gamma }-4{\delta }_{0\alpha }{\delta }_{0\beta }{\delta }_{0\gamma }+2i{\epsilon }_{0\alpha \beta \gamma }\\ \mathrm{tr}\left({\sigma }_{\alpha }{\sigma }_{\beta }{\sigma }_{\gamma }{\sigma }_{\mu }\right)& =2\left({\delta }_{\alpha \beta }{\delta }_{\gamma \mu }-{\delta }_{\alpha \gamma }{\delta }_{\beta \mu }+{\delta }_{\alpha \mu }{\delta }_{\beta \gamma }\right)+4\left({\delta }_{\alpha \gamma }{\delta }_{0\beta }{\delta }_{0\mu }+{\delta }_{\beta \mu }{\delta }_{0\alpha }{\delta }_{0\gamma }\right)-8{\delta }_{0\alpha }{\delta }_{0\beta }{\delta }_{0\gamma }{\delta }_{0\mu }+2i\sum _{(\alpha \beta \gamma \mu )}{\epsilon }_{0\alpha \beta \gamma }{\delta }_{0\mu }\end{array}}$$

where Greek indices α, β, γ and μ assume values from {0, x, y, z} and the notation $\sum _{(\alpha \dots )}$ is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector

For

${\displaystyle \overrightarrow{a}=a\hat{n},|\hat{n}|=1,}$

one has, for even powers, 2p, p = 0, 1, 2, 3, ...

${\displaystyle (\hat{n}\cdot \overrightarrow{\sigma }{)}^{2p}=I}$

which can be shown first for the p = 1 case using the anticommutation relations. For convenience, the case p = 0 is taken to be I by convention.

For odd powers, 2q + 1, q = 0, 1, 2, 3, ...

${\displaystyle {\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}^{2q+1}=\hat{n}\cdot \overrightarrow{\sigma }.}$

Matrix exponentiating, and using the Taylor series for sine and cosine,

${\displaystyle \begin{array}{rl}{e}^{ia\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}& =\sum _{k=0}^{\mathrm{\infty }}\frac{i^k{\left[a\left(\hat{n}\cdot \overrightarrow{\sigma }\right)\right]}^k}{k!}\\ & =\sum _{p=0}^{\mathrm{\infty }}\frac{(-1{)}^p(a\hat{n}\cdot \overrightarrow{\sigma }{)}^{2p}}{(2p)!}+i\sum _{q=0}^{\mathrm{\infty }}\frac{(-1{)}^q(a\hat{n}\cdot \overrightarrow{\sigma }{)}^{2q+1}}{(2q+1)!}\\ & =I\sum _{p=0}^{\mathrm{\infty }}\frac{(-1{)}^p{a}^{2p}}{(2p)!}+i(\hat{n}\cdot \overrightarrow{\sigma })\sum _{q=0}^{\mathrm{\infty }}\frac{(-1{)}^q{a}^{2q+1}}{(2q+1)!}\end{array}}$.

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,

${\displaystyle e^{ia\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}=I\cos a+i(\hat{n}\cdot \overrightarrow{\sigma })\sin a}$

(2)

which is analogous to Euler's formula, extended to quaternions.

Note that

${\displaystyle det[ia(\hat{n}\cdot \overrightarrow{\sigma })]=a^2}$,

while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).

A more abstract version of formula (2) for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of (2) for an analytic (at a and −a) function is provided by application of Sylvester's formula,

${\displaystyle f(a(\hat{n}\cdot \overrightarrow{\sigma }))=I\frac{f(a)+f(-a)}{2}+\hat{n}\cdot \overrightarrow{\sigma }\frac{f(a)-f(-a)}{2}.}$

The group composition law of SU(2)

A straightforward application of formula (2) provides a parameterization of the composition law of the group SU(2). One may directly solve for c in

$${\displaystyle \begin{array}{rl}{e}^{ia\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}{e}^{ib\left(\hat{m}\cdot \overrightarrow{\sigma }\right)}& =I\left(\cos a\cos b-\hat{n}\cdot \hat{m}\sin a\sin b\right)+i\left(\hat{n}\sin a\cos b+\hat{m}\sin b\cos a-\hat{n}\times \hat{m}\sin a\sin b\right)\cdot \overrightarrow{\sigma }\\ & =I\cos c+i\left(\hat{k}\cdot \overrightarrow{\sigma }\right)\sin c\\ & =e^{ic\left(\hat{k}\cdot \overrightarrow{\sigma }\right)},\end{array}}$$

which specifies the generic group multiplication, where, manifestly,

$${\displaystyle \cos c=\cos a\cos b-\hat{n}\cdot \hat{m}\sin a\sin b,}$$

the spherical law of cosines. Given c, then,

$${\displaystyle \hat{k}=\frac{1}{\sin c}\left(\hat{n}\sin a\cos b+\hat{m}\sin b\cos a-\hat{n}\times \hat{m}\sin a\sin b\right).}$$

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to

$${\displaystyle e^{ic\hat{k}\cdot \overrightarrow{\sigma }}=\exp \left(i\frac{c}{\sin c}\left(\hat{n}\sin a\cos b+\hat{m}\sin b\cos a-\hat{n}\times \hat{m}\sin a\sin b\right)\cdot \overrightarrow{\sigma }\right).}$$

(Of course, when ${\displaystyle \hat{n}}$ is parallel to ${\displaystyle \hat{m}}$, so is ${\displaystyle \hat{k}}$, and c = a + b.)

See also: Rotation formalisms in three dimensions § Rodrigues vector, and Spinor § Three dimensions

Adjoint action

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle ${\displaystyle a}$ along any axis ${\displaystyle \hat{n}}$:

$${\displaystyle R_n(-a)\overrightarrow{\sigma }{R}_n(a)=e^{i\frac{a}{2}\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}\overrightarrow{\sigma }{e}^{-i\frac{a}{2}\left(\hat{n}\cdot \overrightarrow{\sigma }\right)}=\overrightarrow{\sigma }\cos (a)+\hat{n}\times \overrightarrow{\sigma }\sin (a)+\hat{n}\hat{n}\cdot \overrightarrow{\sigma }(1-\cos (a)).}$$

Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example, it can be shown that $R_y\left(-\frac{\pi }{2}\right){\sigma }_x{R}_y\left(\frac{\pi }{2}\right)=\hat{x}\cdot \left(\hat{y}\times \overrightarrow{\sigma }\right)={\sigma }_z$.

See also: Rodrigues' rotation formula

Completeness relation

An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the k-th Pauli matrix is σ ^k_αβ.

In this notation, the completeness relation for the Pauli matrices can be written

${\displaystyle {\overrightarrow{\sigma }}_{\alpha \beta }\cdot {\overrightarrow{\sigma }}_{\gamma \delta }\equiv \sum _{k=1}^3{\sigma }_{\alpha \beta }^k{\sigma }_{\gamma \delta }^k=2{\delta }_{\alpha \delta }{\delta }_{\beta \gamma }-{\delta }_{\alpha \beta }{\delta }_{\gamma \delta }.}$

Proof

The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the Hilbert space of all 2 × 2 complex matrices means that we can express any matrix M as

$${\displaystyle M=cI+\sum _k{a}_k{\sigma }^k}$$

where c is a complex number, and a is a 3-component, complex vector. It is straightforward to show, using the properties listed above, that

$${\displaystyle \mathrm{tr}\left({\sigma }^j{\sigma }^k\right)=2{\delta }_{jk}}$$

where "tr" denotes the trace, and hence that

$${\displaystyle \begin{array}{rl}c& =\frac{1}{2}\mathrm{tr}M,\begin{array}{rlrl}& & a_k& =\frac{1}{2}\mathrm{tr}{\sigma }^{k}M.\end{array}\\ \therefore 2M& =I\mathrm{tr}M+\sum _k{\sigma }^k\mathrm{tr}{\sigma }^{k}M,\end{array}}$$

which can be rewritten in terms of matrix indices as

$${\displaystyle 2M_{\alpha \beta }={\delta }_{\alpha \beta }{M}_{\gamma \gamma }+\sum _k{\sigma }_{\alpha \beta }^k{\sigma }_{\gamma \delta }^k{M}_{\delta \gamma },}$$

where summation over the repeated indices is implied γ and δ. Since this is true for any choice of the matrix M, the completeness relation follows as stated above. Q.E.D.

As noted above, it is common to denote the 2 × 2 unit matrix by σ₀, so σ⁰_αβ = δ_αβ. The completeness relation can alternatively be expressed as

$${\displaystyle \sum _{k=0}^3{\sigma }_{\alpha \beta }^k{\sigma }_{\gamma \delta }^k=2{\delta }_{\alpha \delta }{\delta }_{\beta \gamma }.}$$

The fact that any Hermitian complex 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states’ density matrix, (positive semidefinite 2 × 2 matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of {σ₀, σ₁, σ₂, σ₃} as above, and then imposing the positive-semidefinite and trace 1 conditions.

For a pure state, in polar coordinates,

$${\displaystyle \overrightarrow{a}=\left(\begin{array}{ccc}\sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \end{array}\right),}$$

the idempotent density matrix

$${\displaystyle \frac{1}{2}\left(\mathbf{1}+\overrightarrow{a}\cdot \overrightarrow{\sigma }\right)=\left(\begin{array}{cc}{\cos }^2\left(\frac{\theta }{2}\right)& e^{-i\varphi }\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)\\ e^{+i\varphi }\sin \left(\frac{\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)& {\sin }^2\left(\frac{\theta }{2}\right)\end{array}\right)}$$

acts on the state eigenvector ${\displaystyle \left(\begin{array}{cc}\cos \left(\frac{\theta }{2}\right)& e^{+i\varphi }\sin \left(\frac{\theta }{2}\right)\end{array}\right)}$ with eigenvalue +1, hence it acts like a projection operator.

Relation with the permutation operator

Let P_jk be the transposition (also known as a permutation) between two spins σ_j and σ_k living in the tensor product space ${\displaystyle {\mathbb{C}}^2\otimes {\mathbb{C}}^2}$,

${\displaystyle P_{jk}|{\sigma }_j{\sigma }_k⟩=|{\sigma }_k{\sigma }_j⟩.}$

This operator can also be written more explicitly as Dirac's spin exchange operator,

${\displaystyle P_{jk}=\frac{1}{2}\left({\overrightarrow{\sigma }}_j\cdot {\overrightarrow{\sigma }}_k+1\right).}$

Its eigenvalues are therefore^[d] 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.

SU(2)

The group SU(2) is the Lie group of unitary 2 × 2 matrices with unit determinant; its Lie algebra is the set of all 2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra ${\displaystyle {\mathfrak{s}\mathfrak{u}}_2}$ is the 3-dimensional real algebra spanned by the set {iσ_k}. In compact notation,

${\displaystyle \mathfrak{s}\mathfrak{u}(2)=\mathrm{span}\{i{\sigma }_1,i{\sigma }_2,i{\sigma }_3\}.}$

As a result, each iσ_j can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2, so that

${\displaystyle \mathfrak{s}\mathfrak{u}(2)=\mathrm{span}\left\{\frac{i{\sigma }_1}{2},\frac{i{\sigma }_2}{2},\frac{i{\sigma }_3}{2}\right\}.}$

As SU(2) is a compact group, its Cartan decomposition is trivial.

SO(3)

The Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$ is isomorphic to the Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$, which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that the iσ_j are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though ${\displaystyle \mathfrak{s}\mathfrak{u}(2)}$ and ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one group homomorphism from SU(2) to SO(3), see relationship between SO(3) and SU(2).

Quaternions

Main article: Spinor § Three dimensions

The real linear span of {I, iσ₁, iσ₂, iσ₃} is isomorphic to the real algebra of quaternions, ${\displaystyle \mathbb{H}}$, represented by the span of the basis vectors ${\displaystyle \left\{\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}\right\}.}$ The isomorphism from ${\displaystyle \mathbb{H}}$ to this set is given by the following map (notice the reversed signs for the Pauli matrices):

$${\displaystyle \mathbf{1}\mapsto I,\mathbf{i}\mapsto -{\sigma }_2{\sigma }_3=-i{\sigma }_1,\mathbf{j}\mapsto -{\sigma }_3{\sigma }_1=-i{\sigma }_2,\mathbf{k}\mapsto -{\sigma }_1{\sigma }_2=-i{\sigma }_3.}$$

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,

${\displaystyle \mathbf{1}\mapsto I,\mathbf{i}\mapsto i{\sigma }_3,\mathbf{j}\mapsto i{\sigma }_2,\mathbf{k}\mapsto i{\sigma }_1.}$

As the set of versors U ⊂ ${\displaystyle \mathbb{H}}$ forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The two-to-one homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.

Physics

Classical mechanics

Main article: Quaternions and spatial rotation

In classical mechanics, Pauli matrices are useful in the context of the Cayley-Klein parameters. The matrix P corresponding to the position ${\displaystyle \overrightarrow{x}}$ of a point in space is defined in terms of the above Pauli vector matrix,

${\displaystyle P=\overrightarrow{x}\cdot \overrightarrow{\sigma }=x{\sigma }_x+y{\sigma }_y+z{\sigma }_z.}$

Consequently, the transformation matrix Q_θ for rotations about the x-axis through an angle θ may be written in terms of Pauli matrices and the unit matrix as

${\displaystyle Q_{\theta }=\mathbf{1}\cos \frac{\theta }{2}+i{\sigma }_x\sin \frac{\theta }{2}.}$

Similar expressions follow for general Pauli vector rotations as detailed above.

Quantum mechanics

In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin 1⁄2 particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above, iσ_j are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin 1⁄2. The states of the particles are represented as two-component spinors. In the same way, the Pauli matrices are related to the isospin operator.

An interesting property of spin 1⁄2 particles is that they must be rotated by an angle of 4π in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north–south pole on the 2-sphere S², they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.

For a spin 1⁄2 particle, the spin operator is given by J = ħ/2σ, the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli matrices.

Relativistic quantum mechanics

In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as

${\displaystyle {\mathsf{\Sigma }}_k=\left(\begin{array}{cc}{\sigma }_k& 0\\ 0& {\sigma }_k\end{array}\right).}$

It follows from this definition that the ${\displaystyle {\mathsf{\Sigma }}_k}$ matrices have the same algebraic properties as the σ_k matrices.

However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence ${\displaystyle {\mathsf{\Sigma }}_k}$ needs to be replaced by Σ_μν , the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σ_μν are also antisymmetric. Hence there are only six independent matrices.

The first three are the ${\displaystyle {\mathrm{\Sigma }}_{k\ell }\equiv {ϵ}_{jk\ell }{\mathsf{\Sigma }}_j.}$ The remaining three, ${\displaystyle -i{\mathrm{\Sigma }}_{0k}\equiv {\alpha }_k,}$ where the Dirac α_k matrices are defined as

${\displaystyle {\alpha }_k=\left(\begin{array}{cc}0& {\sigma }_k\\ {\sigma }_k& 0\end{array}\right).}$

The relativistic spin matrices Σ_μν are written in compact form in terms of commutator of gamma matrices as

${\displaystyle {\mathrm{\Sigma }}_{\mu \nu }=\frac{i}{2}[{\gamma }_{\mu },{\gamma }_{\nu }].}$

Quantum information

In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the "Z–Y decomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar "X–Y decomposition of a single-qubit gate.

Biopsychosocial model

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Biopsychosocial_model

The biopsychosocial model of health

Biopsychosocial models are a class of trans-disciplinary models which look at the interconnection between biology, psychology, and socio-environmental factors. These models specifically examine how these aspects play a role in topics ranging from human development, to health and disease, to information processing, and to conflict.

According to Derick T. Wade and Peter W. Halligan, as of 2017, it is generally accepted that "illness and health are the result of an interaction between biological, psychological, and social factors."

The term was first used to describe a model advocated by George L. Engel in 1977. The idea behind the model was to express mental distress as a triggered response of a disease that a person is genetically vulnerable when stressful life events occur. In that sense, it is also known as vulnerability-stress model. It is now referred to as a generalized model that interprets similar aspects, and has become an alternative to the biomedical and/or psychological dominance of many health care systems.

History

George L. Engel and Jon Romano of the University of Rochester in 1977, are widely credited with being the first to propose a biopsychosocial model. However, it had been proposed 100 years earlier and by others. Engel struggled with the then-prevailing biomedical approach to medicine as he strove for a more holistic approach by recognizing that each patient has their own thoughts, feelings, and history. In developing his model, Engel framed it for both illnesses and psychological problems.

Engel's biopsychosocial model reflects the development of illness through the complex interaction of biological factors (genetic, biochemical, etc.), psychological factors (mood, personality, behaviour, etc.) and social factors (cultural, familial, socioeconomic, medical, etc.). For example, a person may have a genetic predisposition for depression, but they must have social factors such as extreme stress at work and family life and psychological factors such as a perfectionistic tendencies which all trigger this genetic code for depression. A person may have a genetic predisposition for a disease, but social and cognitive factors must trigger the illness.

Specifically, Engel broadened medical thinking by re-proposing a separation of body and mind. The idea of mind–body dualism goes back at least to René Descartes, but was forgotten during the biomedical approach. Engel emphasized that the biomedical approach is flawed because the body alone does not contribute to illness. Instead, the individual mind (psychological and social factors) play a significant role in how an illness is caused and how it is treated. Engel proposed a dialogue between the patient and the doctor in order to find the most effective treatment solution.

Similarly, materialistic and reductionist ideas proposed with the biomedical model are flawed because they cannot be verified on a cellular level (according to Engel). Instead, the proposed model focuses on the research of past psychologists such as Urie Bronfenbrenner, popularized by his belief that social factors play a role in developing illnesses and behaviors. Simply, Engel used Bronfenbrenner's research as a column of his biopsychosocial model and framed this model to display health at the center of social, psychological, and biological aspects.

After publication, the biopsychosocial model was adopted by the World Health Organization (WHO) in 2002 as a basis for the International Classification of Function (ICF). However, The WHO definition of health adopted in 1948 clearly implied a broad socio-medical perspective.

Current status of the BPS model

The biopsychosocial (BPS) model is still widely used as both a philosophy of clinical care and a practical clinical guide useful for broadening the scope of a clinician's gaze. Dr. Borrell-Carrió and colleagues reviewed Engel's model 25 years on. They proposed the model had evolved into a BPS and relationship-centered framework for physicians. They proposed three clarifications to the model, and identified seven established principles.

1. Self-awareness.
2. Active cultivation of trust.
3. An emotional style characterized by empathic curiosity.
4. Self-calibration as a way to reduce bias.
5. Educating the emotions to assist with diagnosis and forming therapeutic relationships.
6. Using informed intuition.
7. Communicating clinical evidence to foster dialogue, not just the mechanical application of protocol.

Gatchel and colleagues argued in 2007 the biopsychosocial model is the most widely accepted as the most heuristic approach to understanding and treating chronic pain.

Relevant theories and theorists

Other theorists and researchers are using the term biopsychosocial, or sometimes bio-psycho-social to distinguish Engel's model.

Lumley and colleagues used a non-Engel model to conduct a biopsychosocial assessment of the relationship between and pain and emotion. Zucker and Gomberg used a non-Engel biopsychosocial perspective to assess the etiology of alcoholism in 1986.

Crittenden considers the Dynamic-Maturational Model of Attachment and Adaptation (DMM), to be a biopsychosocial model. It incorporates many disciplines to understand human development and information processing.

Kozlowska's Functional Somatic Symptoms model uses a biopsychosocial approach to understand somatic symptoms. Siegel's Interpersonal Neurobiology (IPNB) model is similar, although, perhaps to distinguish IPNB from Engel's model, he describes how the brain, mind, and relationships are part of one reality rather three separate elements. Most trauma -and violence-informed care models are biopsychosocial models.

Biopsychosocial research

Wickrama and colleagues have conducted several biopsychosocial-based studies examining marital dynamics. In a longitudinal study of women divorced midlife they found that divorce contributed to an adverse biopsychosocial process for the women. In another study of enduring marriages, they looked to see if hostile marital interactions in the early middle years could wear down couples regulator systems through greater psychological distress, more health-risk behaviors, and a higher body mass index (BMI). Their findings confirmed negative outcomes and increased vulnerability to later physical health problems for both husbands and wives.

Kovacs and colleagues meta-study examined the biopsychosocial experiences of adults with congenital heart disease. Zhang and colleagues used a biopsychosocial approach to examine parents own physiological response when facing children's negative emotions, and how it related to parents’ ability to engage in sensitive and supportive behaviors. They found parents’ physiological regulatory functioning was an important factor in shaping parenting behaviors directed toward children's emotions.

A biopsychosocial approach was used to assess race and ethnic differences in aging and to develop the Michigan Cognitive Aging Project. Banerjee and colleagues used a biopsychosocial narrative to describe the dual pandemic of suicide and COVID-19.

Potential applications

When Engel first proposed the biopsychosocial model it was for the purpose of better understanding health and illness. While this application still holds true the model is relevant to topics such as health, medicine, and development. Firstly, as proposed by Engel, it helps physicians better understand their whole patient. Considering not only physiological and medical aspects but also psychological and sociological well-being. Furthermore, this model is closely tied to health psychology. Health psychology examines the reciprocal influences of biology, psychology, behavioral, and social factors on health and illness.

One application of the biopsychosocial model within health and medicine relates to pain, such that several factors outside an individual's health may affect their perception of pain. For example, a 2019 study linked genetic and biopsychosocial factors to increased post-operative shoulder pain. Future studies are needed to model and further explore the relationship between biopsychosocial factors and pain.

The developmental applications of this model are equally relevant. One particular advantage of applying the biopsychosocial model to developmental psychology is that it allows for an intersection within the nature versus nurture debate. This model provides developmental psychologists a theoretical basis for the interplay of both hereditary and psychosocial factors on an individual's development.

In gender

Gender is thought by some as biopsychosocial, they define it as a complex matter formed from social, psychological, and biological aspects.

According to the Gender Spectrum Organization, "A person's gender is the complex interrelationship between three dimensions: body, identity, and social gender."

According to Alex Iantaffi and Meg-John Barker, biological, psychological, and social factors all feed back into each other in complex ways to form a person's gender.

Criticisms

There have been a number of criticisms of Engel's biopsychosocial model. Benning summarized the arguments against the model including that it lacked philosophical coherence, was insensitive to patients' subjective experience, was unfaithful to the general systems theory that Engel claimed it be rooted in, and that it engendered an undisciplined eclecticism that provides no safeguards against either the dominance or the under-representation of any one of the three domains of bio, psycho, or social.

Psychiatrist Hamid Tavakoli argues that Engel's BPS model should be avoided because it unintentionally promotes an artificial distinction between biology and psychology, and merely causes confusion in psychiatric assessments and training programs, and that ultimately it has not helped the cause of trying to de-stigmatize mental health. The perspectives model does not make that arbitrary distinction.

A number of these criticisms have been addressed over recent years. For example, the BPS-Pathways model describes how it is possible to conceptually separate, define, and measure biological, psychological, and social factors, and thereby seek detailed interrelationships among these factors.

While Engel's call to arms for a biopsychosocial model has been taken up in several healthcare fields and developed in related models, it has not been adopted in acute medical and surgical domains, as of 2017.

Internal medicine

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Internal_medicine

Internal medicine, also known as general internal medicine in Commonwealth nations, is a medical specialty for medical doctors focused on the prevention, diagnosis, and treatment of internal diseases. Medical practitioners of internal medicine are referred to as internists, or physicians in Commonwealth nations. Internists possess specialized skills in managing patients with undifferentiated or multi-system disease processes. They provide care to both hospitalized (inpatient) and ambulatory (outpatient) patients and often contribute significantly to teaching and research. Internists are qualified physicians who have undergone postgraduate training in internal medicine, and should not be confused with "interns”, a term commonly used for a medical doctor who has obtained a medical degree but does not yet have a license to practice medicine unsupervised.

In the United States and Commonwealth nations, there is often confusion between internal medicine and family medicine, with people mistakenly considering them equivalent.

Internists primarily work in hospitals, as their patients are frequently seriously ill or require extensive medical tests. Internists often have subspecialty interests in diseases affecting particular organs or organ systems. The certification process and available subspecialties may vary across different countries.

Additionally, internal medicine is recognized as a specialty within clinical pharmacy and veterinary medicine.

Etymology and historical development

Robert Koch, 19th century German physician and microbiologist

The term internal medicine in English has its etymology in the 19th-century German term Innere Medizin. Originally, internal medicine focused on determining the underlying "internal" or pathological causes of symptoms and syndromes through a combination of medical tests and bedside clinical examination of patients. This approach differed from earlier generations of physicians, such as the 17th-century English physician Thomas Sydenham, known as the father of English medicine or "the English Hippocrates." Sydenham developed the field of nosology (the study of diseases) through a clinical approach that involved diagnosing and managing diseases based on careful bedside observation of the natural history of disease and their treatment. Sydenham emphasized understanding the internal mechanisms and causes of symptoms rather than dissecting cadavers and scrutinizing the internal workings of the body.

In the 17th century, there was a shift towards anatomical pathology and laboratory studies, and Giovanni Battista Morgagni, an Italian anatomist of the 18th century, is considered the father of anatomical pathology. Laboratory investigations gained increasing significance, with contributions from physicians like German physician and bacteriologist Robert Koch in the 19th century. During this time, internal medicine emerged as a field that integrated the clinical approach with the use of investigations. Many American physicians of the early 20th century studied medicine in Germany and introduced this medical field to the United States, adopting the name "internal medicine" in imitation of the existing German term.

Internal medicine has historical roots in ancient India and ancient China. The earliest texts about internal medicine can be found in the Ayurvedic anthologies of Charaka.

Role of internal medicine specialists

Internal medicine specialists, also referred to as general internal medicine specialists or general medicine physicians in Commonwealth countries, are specialized physicians trained to manage complex or multisystem disease conditions that single-organ specialists may not be equipped to handle. They are often called upon to address undifferentiated presentations that do not fit neatly within the scope of a single-organ specialty, such as shortness of breath, fatigue, weight loss, chest pain, confusion, or alterations in conscious state. They may manage serious acute illnesses that affect multiple organ systems concurrently within a single patient, as well as the management of multiple chronic diseases in a single patient.

While many internal medicine physicians choose to subspecialize in specific organ systems, general internal medicine specialists do not necessarily possess any lesser expertise than single-organ specialists. Rather, they are specifically trained to care for patients with multiple simultaneous problems or complex comorbidities.

Due to the complexity involved in explaining the treatment of diseases that are not localized to a single organ, there has been some confusion surrounding the meaning of internal medicine and the role of an "internist." Although internists may serve as primary care physicians, they are not synonymous with "family physicians," "family practitioners," "general practitioners," or "GPs." The training of internists is solely focused on adults and does not typically include surgery, obstetrics, or pediatrics. According to the American College of Physicians, internists are defined as "physicians who specialize in the prevention, detection, and treatment of illnesses in adults." While there may be some overlap in the patient population served by both internal medicine and family medicine physicians, internists primarily focus on adult care with an emphasis on diagnosis, whereas family medicine incorporates a holistic approach to care for the entire family unit. Internists also receive substantial training in various recognized subspecialties within the field and are experienced in both inpatient and outpatient settings. On the other hand, family medicine physicians receive education covering a wide range of conditions and typically train in an outpatient setting with less exposure to hospital settings. The historical roots of internal medicine can be traced back to the incorporation of scientific principles into medical practice in the 1800s, while family medicine emerged as part of the primary care movement in the 1960s.

Education and training

Main article: Medical education

The training and career pathways for internists vary considerably across different countries.

Many programs require previous undergraduate education prior to medical school admission. This "pre-medical" education is typically four or five years in length. Graduate medical education programs vary in length by country. Medical education programs are tertiary-level courses, undertaken at a medical school attached to a university. In the US, medical school consists of four years. Hence, gaining a basic medical education may typically take eight years, depending on jurisdiction and university.

Following completion of entry-level training, newly graduated medical practitioners are often required to undertake a period of supervised practice before their licensure, or registration, is granted, typically one or two years. This period may be referred to as "internship", "conditional registration", or "foundation programme". Then, doctors may follow specialty training in internal medicine if they wish, typically being selected to training programs through competition. In North America, this period of postgraduate training is referred to as residency training, followed by an optional fellowship if the internist decides to train in a subspecialty.

In most countries, residency training for internal medicine lasts three years and centers on secondary and tertiary levels of health care, as opposed to primary health care. In Commonwealth countries, trainees are often called senior house officers for four years after the completion of their medical degree (foundation and core years). After this period, they are able to advance to registrar grade when they undergo a compulsory subspecialty training (including acute internal medicine or a dual subspecialty including internal medicine). This latter stage of training is achieved through competition rather than just by yearly progress as the first years of postgraduate training.

Certification

In the US, three organizations are responsible for the certification of trained internists (i.e., doctors who have completed an accredited residency training program) in terms of their knowledge, skills, and attitudes that are essential for patient care: the American Board of Internal Medicine, the American Osteopathic Board of Internal Medicine and the Board of Certification in Internal Medicine. In the UK, the General Medical Council oversees licensing and certification of internal medicine physicians. The Royal Australasian College of Physicians confers fellowship to internists (and sub-specialists) in Australia. The Medical Council of Canada oversees licensing of internists in Canada.

Subspecialties

United States of America

In the US, two organizations are responsible for certification of subspecialists within the field: the American Board of Internal Medicine and the American Osteopathic Board of Internal Medicine. Physicians (not only internists) who successfully pass board exams receive "board certified" status.

American Board of Internal Medicine

The following are the subspecialties recognized by the American Board of Internal Medicine.

• Adolescent medicine
• Adult congenital heart disease
• Advanced heart failure and transplant cardiology
• Allergy and immunology, concerned with the diagnosis, treatment and management of allergies, asthma and disorders of the immune system.
• Cardiovascular disease, dealing with disorders of the heart and blood vessels*
• Clinical cardiac electrophysiology
• Critical care medicine
• Endocrinology, diabetes & metabolism, dealing with disorders of the endocrine system and its specific secretions called hormones
• Gastroenterology, concerned with the field of digestive diseases
• Geriatric medicine
• Hematology, concerned with blood, the blood-forming organs and its disorders.
• Hospice & palliative medicine
• Infectious disease, concerned with disease caused by a biological agent such as by a virus, bacterium or parasite
• Interventional cardiology
• Medical oncology, dealing with the chemotherapeutic (chemical) and/or immunotherapeutic (immunological) treatment of cancer
• Nephrology, dealing with the study of the function and diseases of the kidney
• Neurocritical care
• Pulmonary disease, dealing with diseases of the lungs and the respiratory tract
• Rheumatology, devoted to the diagnosis and therapy of rheumatic diseases
• Sleep medicine
• Sports medicine
• Transplant hepatology

American College of Osteopathic Internists

The American College of Osteopathic Internists recognizes the following subspecialties:

• Allergy/immunology
• Cardiology
• Cardiac electrophysiology
• Critical care medicine
• Endocrinology
• Gastroenterology
• Geriatrics
• Hematology/oncology
• Interventional cardiology
• Infectious diseases
• Nephrology
• Oncology
• Palliative care medicine
• Pulmonary Diseases
• Pulmonology
• Rheumatology
• Sleep medicine

United Kingdom

In the United Kingdom, the three medical Royal Colleges (the Royal College of Physicians of London, the Royal College of Physicians of Edinburgh and the Royal College of Physicians and Surgeons of Glasgow) are responsible for setting curricula and training programmes through the Joint Royal Colleges Postgraduate Training Board (JRCPTB), although the process is monitored and accredited by the independent General Medical Council (which also maintains the specialist register).

Doctors who have completed medical school spend two years in foundation training completing a basic postgraduate curriculum. After two years of Core Medical Training (CT1/CT2), or three years of Internal Medicine Training (IMT1/IMT2/IMT3) as of 2019, since and attaining the Membership of the Royal College of Physicians, physicians commit to one of the medical specialties:

• Acute internal medicine (with possible subspecialty in stroke medicine)
• Allergy
• Audio vestibular medicine
• Aviation and space medicine
• Cardiology (with possible subspecialty in stroke medicine)
• Clinical genetics
• Clinical neurophysiology
• Clinical oncology
• Clinical pharmacology and therapeutics (with possible subspecialty in stroke medicine)
• Dermatology
• Endocrinology and diabetes mellitus
• Gastroenterology (with possible subspecialty in hepatology)
• General (internal) medicine (with possible subspecialty in metabolic medicine or stroke medicine)
• Genito-urinary medicine
• Geriatric medicine (with possible subspecialty in stroke medicine)
• Haematology
• Immunology
• Infectious diseases
• Intensive care medicine
• Medical microbiology
• Medical oncology (clinical or radiation oncology falls under the Royal College of Radiologists, although entry is through CMT and MRCP is required)
• Medical ophthalmology
• Medical virology
• Neurology (with possible subspecialty in stroke medicine)
• Nuclear medicine
• Occupational medicine
• Paediatric cardiology (the only pediatric subspecialty not under the Royal College of Paediatrics and Child Health)
• Palliative medicine
• Rehabilitation medicine (with possible subspecialty in stroke medicine)
• Renal medicine
• Respiratory medicine
• Rheumatology
• Sport and exercise medicine
• Tropical medicine

Many training programmes provide dual accreditation with general (internal) medicine and are involved in the general care to hospitalised patients. These are acute medicine, cardiology, Clinical Pharmacology and Therapeutics, endocrinology and diabetes mellitus, gastroenterology, infectious diseases, renal medicine, respiratory medicine and often, rheumatology. The role of general medicine, after a period of decline, was reemphasised by the Royal College of Physicians of London report from the Future Hospital Commission (2013).

European Union

The European Board of Internal Medicine (EBIM) was formed as a collaborative effort between the European Union of Medical Specialists (UEMS) - Internal Medicine Section and the European Federation of Internal Medicine (EFIM) to provide guidance on standardizing training and practice of internal medicine throughout Europe. The EBIM published training requirements in 2016 for postgraduate education in internal medicine, and efforts to create a European Certificate of Internal Medicine (ECIM) to facilitate the free movement of medical professionals with the EU are currently underway.

The internal medicine specialist is recognized in every country in the European Union and typically requires five years of multi-disciplinary post-graduate education. The specialty of internal medicine is seen as providing care in a wide variety of conditions involving every organ system and is distinguished from family medicine in that the latter provides a broader model of care the includes both surgery and obstetrics in both adults and children.

Australia

Accreditation for medical education and training programs in Australia is provided by the Australian Medical Council (AMC) and the Medical Council of New Zealeand (MCNZ). The Medical Board of Australia (MBA) is the registering body for Australian doctors and provides information to the Australian Health Practitioner Regulation Agency (AHPRA). Medical graduates apply for provisional registration in order to complete intern training. Those completing an accredited internship program are then eligible to apply for general registration. Once the candidate completes the required basic and advanced post-graduate training and a written and clinical examination, the Royal Australasian College of Physicians confers designation Fellow of the Royal Australasian College of Physicians (FRACP). Basic training consists of three years of full-time equivalent (FTE) training (including intern year) and advanced training consists of 3–4 years, depending on specialty. The fields of specialty practice are approved by the Council of Australian Governments (COAG) and managed by the MBA. The following is a list of currently recognized specialist physicians.

• Cardiology
• Clinical genetics
• Clinical pharmacology
• Endocrinology
• Gastroenterology and hepatology
• General medicine
• Geriatric medicine
• Haemotology
• Immunology and allergy
• Infectious diseases
• Medical oncology
• Nephrology
• Neurology
• Nuclear medicine
• Respiratory and sleep medicine
• Rheumatology

Canada

After completing medical school, internists in Canada require an additional four years of training. Internists desiring to subspecialize are required to complete two additional years of training that may begin after the third year of internist training. The Royal College of Physicians and Surgeons of Canada (RCPSC) is a national non-profit agency that oversees and accredits medical education in Canada. A full medical license in Internal Medicine in Canada requires a medical degree, a license from the Medical Council of Canada, completion of the required post-graduate education, and certification from the RCPSC. Any additional requirements from separate medical regulatory authorities in each province or territory is also required. Internists may practice in Canada as generalists in Internal Medicine or serve in one of seventeen subspecialty areas. Internists may work in many settings including outpatient clinics, inpatient wards, critical care units, and emergency departments. The currently recognized subspecialties include the following:

• Critical care medicine
• Cardiology
• Infectious diseases
• Neurology
• Respiratory medicine
• Rheumatology
• Endocrinology and metabolism
• Gastroenterology
• General internal medicine
• Geriatrics
• Hematology
• Medical oncology
• Clinical allergy and immunology
• Dermatology

Medical diagnosis and treatment

Medicine is mainly focused on the art of diagnosis and treatment with medication. The diagnostic process involves gathering data, generating one or more diagnostic hypotheses, and iteratively testing these potential diagnoses against dynamic disease profiles to determine the best course of action for the patient.

Gathering data

Data may be gathered directly from the patient in medical history-taking and physical examination. Previous medical records including laboratory findings, imaging, and clinical notes from other physicians is also an important source of information; however, it is vital to talk to and examine the patient to find out what the patient is currently experiencing to make an accurate diagnosis.

History and physical examination are a vital part of the diagnostic process.

Internists often can perform and interpret diagnostic tests like EKGs and ultrasound imaging (Point-of-care Ultrasound – PoCUS).

Internists who pursue sub-specialties have additional diagnostic tools, including those listed below.

• Cardiology: angioplasty, cardioversion, cardiac ablation, intra-aortic balloon pump
• Critical care medicine: mechanical ventilation
• Gastroenterology: endoscopy and ERCP
• Nephrology: dialysis
• Pulmonology: bronchoscopy

Other tests are ordered, and patients are also referred to specialists for further evaluation.  The effectiveness and efficiency of the specialist referral process is an area of potential improvement.

Generating diagnostic hypotheses

Determining which pieces of information are most important to the next phase of the diagnostic process is of vital importance. It is during this stage that clinical bias like anchoring or premature closure may be introduced. Once key findings are determined, they are compared to profiles of possible diseases.  These profiles include findings that are typically associated with the disease and are based on the likelihood that someone with the disease has a particular symptom.  A list of potential diagnoses is termed the “differential diagnosis” for the patient and is typically ordered from most likely to least likely, with special attention given to those conditions that have dire consequences for the patient if they were missed. Epidemiology and endemic conditions are also considered in creating and evaluating the list of diagnoses.

The list is dynamic and changes as the physician obtains additional information that makes a condition more (“rule-in”) or less (“rule-out”) likely based on the disease profile. The list is used to determine what information will be acquired next, including which diagnostic test or imaging modality to order.  The selection of tests is also based on the physician’s knowledge of the specificity and sensitivity of a particular test.

An important part of this process is knowledge of the various ways that a disease can present in a patient.  This knowledge is gathered and shared to add to the database of disease profiles used by physicians. This is especially important in rare diseases.

Communication

Communication is a vital part of the diagnostic process. The Internist uses both synchronous and asynchronous communication with other members of the medical care team, including other internists, radiologists, specialists, and laboratory technicians.  Tools to evaluate teamwork exist and have been employed in multiple settings.

Communication to the patient is also important to ensure there is informed consent and shared decision-making throughout the diagnostic process.

Treatment

Treatment modalities generally include both pharmacological and non-pharmacological, depending on the primary diagnosis. Additional treatment options include referral to specialist care including physical therapy and rehabilitation. Treatment recommendations differ in the acute inpatient and outpatient settings. Continuity of care and long-term follow-up is crucial in successful patient outcomes.

Prevention and other services

Aside from diagnosing and treating acute conditions, the Internist may also assess disease risk and recommend preventive screening and intervention.  Some of the tools available to the Internist include genetic evaluation.

Internists also routinely provide pre-operative medical evaluations including individualized assessment and communication of operative risk.

Training the next generation of internists is an important part of the profession.  As mentioned above, post-graduate medical education is provided by licensed physicians as part of accredited education programs that are usually affiliated with teaching hospitals. Studies show that there are no differences in patient outcomes in teaching versus non-teaching facilities. Medical research is an important part of most post-graduate education programs, and many licensed physicians continue to be involved in research activities after completing post-graduate training.

Ethics

Inherent in any medical profession are legal and ethical considerations. Specific laws vary by jurisdiction and may or may not be congruent with ethical considerations. Thus, a strong ethical foundation is paramount to any medical profession. Medical ethics guidelines in the Western world typically follow four principles including beneficence, non-maleficence, patient autonomy, and justice. These principles underlie the patient-physician relationship and the obligation to put the welfare and interests of the patient above their own.

Patient-physician relationship

The relationship is built upon the physician obligations of competency, respect for the patient, and appropriate referrals while the patient requirements include decision-making and provides or withdraws consent for any treatment plan.  Good communication is key to a strong relationship but has ethical considerations as well, including proper use of electronic communication and clear documentation.

Treatment and telemedicine

Providing treatment including prescribing medications based on remote information gathering without a proper established relationship is not accepted as good practice with few exceptions. These exceptions include cross-coverage within a practice and certain public health urgent or emergent issues.

The ethics of telemedicine including questions on its impact to diagnosis, physician-patient relationship, and continuity of care have been raised; however, with appropriate use and specific guidelines, risks may be minimized and the benefits including increased access to care may be realized.

Financial issues and conflicts of interest

Ethical considerations in financial include accurate billing practices and clearly defined financial relationships.  Physicians have both a professional duty and obligation under the justice principle to ensure that patients are provided the same care regardless of status or ability to pay.  However, informal copayment forgiveness may have legal ramifications and the providing professional courtesy may have negatively impact care.

Physicians must disclose all possible conflicts of interest including financial relationships, investments, research and referral relationships, and any other instances that may subjugate or give the appearance of subjugating patient care to self-interest.

Other topics

Other foundational ethical considerations include privacy, confidentiality, accurate and complete medical records, electronic health records, disclosure, and informed decision-making and consent.

Electronic health records have been shown to improve patient care but have risks including data breaches and inappropriate and/or unauthorized disclosure of protected health information.

Withholding information from a patient is typically seen as unethical and in violation of a patient’s right to make informed decisions.  However, in situations where a patient has requested not to be informed or to have the information provided to a second party or in an emergency situation in which the patient does not have decision-making capacity, withholding information may be appropriate.