Feynman slash notation

Feynman slash notationMain

Community hub

7 pages, 0 posts

0 subscribers

Recent from talks

Be the first to start a discussion here.

Recent from talks

Be the first to start a discussion here.

Contribute something

About hubMembersContent overviewUpdatesRules

Main reference articles

Feynman slash notation

View on Wikipedia

from Wikipedia

In the study of Dirac fields in quantum field theory, Richard Feynman introduced the convenient Feynman slash notation (less commonly known as the Dirac slash notation^[1]). If A is a covariant vector (i.e., a 1-form),

{A\!\!\!/}\ {\stackrel {\mathrm {def} }{=}}\ \gamma ^{0}A_{0}+\gamma ^{1}A_{1}+\gamma ^{2}A_{2}+\gamma ^{3}A_{3}

where γ are the gamma matrices. Using the Einstein summation notation, the expression is simply

{A\!\!\!/}\ {\stackrel {\mathrm {def} }{=}}\ \gamma ^{\mu }A_{\mu }

Identities

[edit]

Using the anticommutators of the gamma matrices, one can show that for any $a_{\mu }$ and $b_{\mu }$ ,

{\begin{aligned}{a\!\!\!/}{a\!\!\!/}=a^{\mu }a_{\mu }\cdot I_{4}=a^{2}\cdot I_{4}\\{a\!\!\!/}{b\!\!\!/}+{b\!\!\!/}{a\!\!\!/}=2a\cdot b\cdot I_{4}.\end{aligned}}

where $I_{4}$ is the identity matrix in four dimensions.

In particular,

{\partial \!\!\!/}^{2}=\partial ^{2}\cdot I_{4}.

Further identities can be read off directly from the gamma matrix identities by replacing the metric tensor with inner products. For example,

{\begin{aligned}\gamma _{\mu }{a\!\!\!/}\gamma ^{\mu }&=-2{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}\gamma ^{\mu }&=4a\cdot b\cdot I_{4}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}\gamma ^{\mu }&=-2{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}\\\gamma _{\mu }{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/}\gamma ^{\mu }&=2({d\!\!\!/}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}+{c\!\!\!/}{b\!\!\!/}{a\!\!\!/}{d\!\!\!/})\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/})&=4a\cdot b\\\operatorname {tr} ({a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&=4\left[(a\cdot b)(c\cdot d)-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)\right]\\\operatorname {tr} ({a\!\!\!/}{\gamma ^{\mu }}{b\!\!\!/}{\gamma ^{\nu }})&=4\left[a^{\mu }b^{\nu }+a^{\nu }b^{\mu }-\eta ^{\mu \nu }(a\cdot b)\right]\\\operatorname {tr} (\gamma _{5}{a\!\!\!/}{b\!\!\!/}{c\!\!\!/}{d\!\!\!/})&=4i\varepsilon _{\mu \nu \lambda \sigma }a^{\mu }b^{\nu }c^{\lambda }d^{\sigma }\\\operatorname {tr} ({\gamma ^{\mu }}{a\!\!\!/}{\gamma ^{\nu }})&=0\\\operatorname {tr} ({\gamma ^{5}}{a\!\!\!/}{b\!\!\!/})&=0\\\operatorname {tr} ({\gamma ^{0}}({a\!\!\!/}+m){\gamma ^{0}}({b\!\!\!/}+m))&=8a^{0}b^{0}-4(a\cdot b)+4m^{2}\\\operatorname {tr} (({a\!\!\!/}+m){\gamma ^{\mu }}({b\!\!\!/}+m){\gamma ^{\nu }})&=4\left[a^{\mu }b^{\nu }+a^{\nu }b^{\mu }-\eta ^{\mu \nu }((a\cdot b)-m^{2})\right]\\\operatorname {tr} ({a\!\!\!/}_{1}...{a\!\!\!/}_{2n})&=\operatorname {tr} ({a\!\!\!/}_{2n}...{a\!\!\!/}_{1})\\\operatorname {tr} ({a\!\!\!/}_{1}...{a\!\!\!/}_{2n+1})&=0\end{aligned}}

where:

$\varepsilon _{\mu \nu \lambda \sigma }$ is the Levi-Civita symbol
$\eta ^{\mu \nu }$ is the Minkowski metric
$m$ is a scalar.

With four-momentum

[edit]

This section uses the $(+ - - -)$ metric signature. Often, when using the Dirac equation and solving for cross sections, one finds the slash notation used on four-momentum: using the Dirac basis for the gamma matrices,

\gamma ^{0}={\begin{pmatrix}I&0\\0&-I\end{pmatrix}},\quad \gamma ^{i}={\begin{pmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\end{pmatrix}}\,

as well as the definition of contravariant four-momentum in natural units,

p^{\mu }=\left(E,p_{x},p_{y},p_{z}\right)\,

we see explicitly that

{\begin{aligned}{p\!\!/}&=\gamma ^{\mu }p_{\mu }=\gamma ^{0}p^{0}-\gamma ^{i}p^{i}\\&={\begin{bmatrix}p^{0}&0\\0&-p^{0}\end{bmatrix}}-{\begin{bmatrix}0&\sigma ^{i}p^{i}\\-\sigma ^{i}p^{i}&0\end{bmatrix}}\\&={\begin{bmatrix}E&-{\vec {\sigma }}\cdot {\vec {p}}\\{\vec {\sigma }}\cdot {\vec {p}}&-E\end{bmatrix}}.\end{aligned}}

Similar results hold in other bases, such as the Weyl basis.

References

[edit]

^ Weinberg, Steven (1995), The Quantum Theory of Fields, vol. 1, Cambridge University Press, p. 358 (380 in polish edition), ISBN 0-521-55001-7

Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.

v t e Richard Feynman
Career	Feynman diagram Feynman–Kac formula Wheeler–Feynman absorber theory Bethe–Feynman formula Hellmann–Feynman theorem Feynman slash notation Feynman parametrization Path integral formulation Parton model Sticky bead argument One-electron universe Quantum cellular automaton Rogers Commission Report Feynman checkerboard Feynman sprinkler
Works	"There's Plenty of Room at the Bottom" (1959) The Feynman Lectures on Physics (1964) The Character of Physical Law (1965) QED: The Strange Theory of Light and Matter (1985) Surely You're Joking, Mr. Feynman! (1985) What Do You Care What Other People Think? (1988) Feynman's Lost Lecture: The Motion of Planets Around the Sun (1997) The Meaning of It All (1999) The Pleasure of Finding Things Out (1999) Perfectly Reasonable Deviations from the Beaten Track (2005)
Family	Joan Feynman (sister) Charles Hirshberg (nephew)
Related	Namesakes One-loop Feynman diagram Quantum Man: Richard Feynman's Life in Science Tuva or Bust! Feynman Prize in Nanotechnology Infinity (1996 film) QED (2001 play) The Challenger Disaster (2013 film)

This quantum mechanics-related article is a stub. You can help Wikipedia by expanding it.

Revisions and contributors Edit on Wikipedia Read on Wikipedia

View on Grokipedia

from Grokipedia

Feynman slash notation, also known as the Dirac slash notation, is a concise mathematical convention in quantum field theory (QFT) that represents the contraction of a four-vector

a^\mu

with the Dirac gamma matrices

\gamma^\mu

, denoted as

\slash{a} = \gamma^\mu a_\mu

. Introduced by physicist Richard Feynman during his development of quantum electrodynamics (QED), this notation simplifies expressions involving spinor fields and is particularly useful for handling the Dirac equation in relativistic contexts. In QFT, the slash notation streamlines calculations for fermion propagators, vertex functions, and scattering amplitudes in Feynman diagrams, where it appears in terms like the electron propagator

i(\slash{p} - m)^{-1}

. Its adoption in standard textbooks underscores its role in making Dirac algebra more tractable, especially when combined with properties of the gamma matrices such as anticommutation relations

\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}

. The notation is metric-dependent, typically using the Minkowski metric

g^{\mu\nu} = \operatorname{diag}(1, -1, -1, -1)

, and extends to other four-vectors like the electromagnetic potential

A^\mu

, yielding

\slash{A}

. Beyond QED, Feynman slash notation is integral to the standard model of particle physics, facilitating computations in weak interactions and QCD processes involving quarks and leptons. Its efficiency has made it a staple in both perturbative and non-perturbative QFT analyses, though care must be taken with conventions for the gamma matrix representation (e.g., Dirac, Weyl, or chiral bases).

Fundamentals

Definition

The Feynman slash notation provides a concise representation for the contraction of a four-vector with the Dirac gamma matrices in the framework of quantum field theory. Specifically, for a covariant four-vector

A_\mu

, the slash operator is defined as

A\!\!/ = \gamma^\mu A_\mu,

where

\gamma^\mu

are the Dirac gamma matrices, the index

\mu

runs over spacetime coordinates (0,1,2,3), and the summation is implied by the Einstein convention. This notation, introduced by Richard Feynman, simplifies expressions involving Dirac fields by avoiding explicit summation symbols. The Dirac gamma matrices

\gamma^\mu

are 4×4 matrices over the complex numbers that generate the Clifford algebra for Minkowski spacetime. In the standard Dirac representation,

\gamma^0

is Hermitian, while the spatial matrices

\gamma^i

(i=1,2,3) are anti-Hermitian.^[1] They satisfy the defining anticommutation relations

\{ \gamma^\mu, \gamma^\nu \} = 2 g^{\mu\nu} I,

where

g^{\mu\nu}

is the Minkowski metric tensor with signature

(+,-,-,-)

, and

I

is the 4×4 identity matrix.^[2] These relations ensure that the gamma matrices encode the Lorentz structure essential for relativistic invariance in fermionic theories. In quantum field theory, the slash notation is employed to describe interactions and propagators of Dirac fields, which model spin-1/2 fermions like quarks and leptons. For instance, applied to a general covariant four-vector

A_\mu

(such as an electromagnetic potential),

A\!\!/

acts as a matrix operator on spinor wave functions, facilitating calculations in covariant perturbation theory.

Notation Conventions

The Feynman slash notation, introduced by Richard Feynman in the context of quantum field theory, is conventionally rendered as

\slash{A}

A\!\!/

, where a diagonal slash is superimposed over the symbol

A

representing a four-vector.^[3] This typographical convention facilitates compact expression of contractions with Dirac gamma matrices, such as

\slash{A} = \gamma^\mu A_\mu

.^[4] Variations in rendering arise to accommodate symbol width and avoid visual overlap; for instance, the momentum four-vector is often denoted as

p\!\!\!/

to ensure the slash aligns properly without crowding the letter.^[5] In printed texts, the notation may appear in boldface for vectors (e.g.,

\mathbf{p}\!\!\!/

) or italics depending on the style guide, though upright symbols are standard for indices.^[6] In LaTeX typesetting, the preferred method employs the \slashed macro from the slashed package, yielding

\slashed{p}

; alternatively, manual adjustment via

p\!\!\!/

achieves similar results for basic documents.^[7] The notation operates within Minkowski spacetime, where metric signature conventions influence index raising and lowering; in particle physics literature, the mostly minus signature

\eta_{\mu\nu} = \operatorname{diag}(+1, -1, -1, -1)

predominates, though the mostly plus

\eta_{\mu\nu} = \operatorname{diag}(-1, +1, +1, +1)

is also employed.^[8]^[9] Care must be taken to distinguish this from slash notation in light-cone coordinates, where a slash denotes null components such as

x^\pm = (t \pm z)/\sqrt{2}

x±=(t±z)/2

.^[10]

Mathematical Properties

Algebraic Identities

The algebraic identities of the Feynman slash notation arise directly from the defining anticommutation relations of the Dirac gamma matrices, which form a representation of the Clifford algebra in four spacetime dimensions. For two arbitrary four-vectors

A^\mu

and

B^\mu

, the slashed operators are defined as

\slashed{A} = A_\mu \gamma^\mu

and

\slashed{B} = B_\nu \gamma^\nu

, where repeated indices imply summation over

\mu, \nu = 0, 1, 2, 3

using the Minkowski metric

\eta^{\mu\nu} = \operatorname{diag}(1, -1, -1, -1)

. The fundamental anticommutator

\{ \gamma^\mu, \gamma^\nu \} = 2 \eta^{\mu\nu} I_4

(with

I_4

the

4 \times 4

identity matrix) leads to the key relation for the product of slashed operators. The anticommutator of two slashed four-vectors satisfies

\{ \slashed{A}, \slashed{B} \} = \slashed{A} \slashed{B} + \slashed{B} \slashed{A} = 2 (A \cdot B) I_4,

where

A \cdot B = A^\mu B_\mu = \eta_{\mu\nu} A^\mu B^\nu

. This follows by expanding:

\slashed{A} \slashed{B} = A_\mu B_\nu \gamma^\mu \gamma^\nu = A_\mu B_\nu \left( \frac{1}{2} \{ \gamma^\mu, \gamma^\nu \} + \frac{1}{2} [ \gamma^\mu, \gamma^\nu ] \right) = A_\mu B_\nu \eta^{\mu\nu} I_4 + \frac{1}{2} A_\mu B_\nu [ \gamma^\mu, \gamma^\nu ],

and similarly for

\slashed{B} \slashed{A}

; the commutator terms cancel in the anticommutator, yielding the scalar multiple of the identity.^[6] [Peskin & Schroeder, 1995, p. 42] A special case is the square of a single slashed vector, which simplifies to

\slashed{A} \slashed{A} = A^2 I_4,

where

A^2 = A^\mu A_\mu

. This is obtained by setting

B = A

in the anticommutator identity, giving

\{ \slashed{A}, \slashed{A} \} = 2 A^2 I_4

, but since

\slashed{A}

commutes with itself,

\slashed{A} \slashed{A} + \slashed{A} \slashed{A} = 2 \slashed{A}^2

, so

\slashed{A}^2 = A^2 I_4

. Alternatively, direct expansion uses

\gamma^\mu \gamma^\nu A_\mu A_\nu = \eta^{\mu\nu} A_\mu A_\nu I_4

for

\mu = \nu

and the antisymmetric commutator vanishing when contracted with symmetric

A_\mu A_\nu

. This identity is crucial for simplifying expressions like the Dirac operator squared in the Klein-Gordon limit.^[6] [p. 84] [Peskin & Schroeder, 1995, p. 43] From the anticommutator, one can rearrange to express the product of distinct slashed operators:

\slashed{A} \slashed{B} = 2 (A \cdot B) I_4 - \slashed{B} \slashed{A}.

This follows immediately by solving

\slashed{A} \slashed{B} = \{ \slashed{A}, \slashed{B} \} - \slashed{B} \slashed{A}

and substituting the anticommutator result; it highlights the non-commutativity of the slash operators unless

A

and

B

are parallel. [Peskin & Schroeder, 1995, p. 42] The commutator form introduces the tensor

\sigma^{\mu\nu}

, defined via the Clifford algebra as

\sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu, \gamma^\nu]

(antisymmetric in

\mu \leftrightarrow \nu

). For slashed vectors, it yields

[ \slashed{A}, \slashed{B} ] = \slashed{A} \slashed{B} - \slashed{B} \slashed{A} = -i \sigma^{\mu\nu} (A_\mu B_\nu - A_\nu B_\mu).

To derive this, expand

[ \slashed{A}, \slashed{B} ] = A_\mu B_\nu [\gamma^\mu, \gamma^\nu]

; since

[\gamma^\mu, \gamma^\nu] = -2i \sigma^{\mu\nu}

from the anticommutator (as

\{ \gamma^\mu, \gamma^\nu \} = 2 \eta^{\mu\nu} I_4

implies the commutator is minus twice the antisymmetric part times i in the definition), the contraction yields

-2i \sigma^{\mu\nu} A_\mu B_\nu

. Since the symmetric part vanishes and

\sigma^{\mu\nu} A_\mu B_\nu = \frac{1}{2} \sigma^{\mu\nu} (A_\mu B_\nu - A_\nu B_\mu)

, this simplifies to

-i \sigma^{\mu\nu} (A_\mu B_\nu - A_\nu B_\mu)

. The

\sigma^{\mu\nu}

tensor generates Lorentz transformations in the spinor representation and appears in magnetic moment interactions.^[6] [p. 78, 115] [Peskin & Schroeder, 1995, pp. 42-43, 116]

Trace Identities

Trace identities involving products of slash operators are crucial for evaluating loop integrals and scattering amplitudes in quantum field theory, as they simplify the computation of traces over Dirac spinor space. These identities exploit the properties of the gamma matrices and the cyclic nature of the trace operation, allowing contraction of Lorentz indices in a compact manner. The simplest nontrivial trace identity is for two slash operators:

\operatorname{Tr}(\slash{A} \slash{B}) = 4 A \cdot B,

where the dot denotes the Lorentz-invariant scalar product, and the trace is taken over the 4-dimensional Dirac space. This follows directly from the basic gamma matrix trace

\operatorname{Tr}(\gamma^\mu \gamma^\nu) = 4 g^{\mu\nu}

.^[6] For an even number of slash operators, the traces can be expanded into sums of scalar products. A representative case is the trace of four slash operators:

\operatorname{Tr}(\slash{A} \slash{B} \slash{C} \slash{D}) = 4 \left[ (A \cdot B)(C \cdot D) - (A \cdot C)(B \cdot D) + (A \cdot D)(B \cdot C) \right].

This identity arises from the general formula for the trace of four gamma matrices,

\operatorname{Tr}(\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma) = 4 (g^{\mu\nu} g^{\rho\sigma} - g^{\mu\rho} g^{\nu\sigma} + g^{\mu\sigma} g^{\nu\rho})

, combined with index contractions. More slashes follow similar patterns but grow combinatorially complex, often requiring recursive algebraic techniques based on the Clifford algebra relations.^[6] Traces involving an odd number of slash operators vanish in four spacetime dimensions,

\operatorname{Tr}(\slash{A}_1 \cdots \slash{A}_{2n+1}) = 0

, due to the antisymmetry of the gamma matrices and the even dimensionality of the Dirac representation, which makes such products traceless. This property simplifies many Feynman diagram evaluations by eliminating certain contributions.^[6] When a pseudoscalar

\gamma_5

is included, traces with an even number of slashes (yielding four gamma matrices total) produce a Levi-Civita tensor structure, reflecting the axial nature of

\gamma_5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3

\operatorname{Tr}(\slash{A} \slash{B} \slash{C} \slash{D} \gamma_5) = 4i \epsilon_{\mu\nu\rho\sigma} A^\mu B^\nu C^\rho D^\sigma,

where

\epsilon_{\mu\nu\rho\sigma}

is the totally antisymmetric tensor with

\epsilon_{0123} = +1

. This identity is essential for processes involving chiral asymmetries, such as weak interactions. Traces with

\gamma_5

and fewer than four slashes vanish, while those with more require reduction using algebraic identities.^[11] In dimensional regularization, where spacetime is continued to

d = 4 - 2\epsilon

dimensions to handle divergences, these trace identities are modified. The trace of the identity becomes

\operatorname{Tr}(1) = 2^{d/2}

, altering prefactors in even-slash traces (e.g., the two-slash identity generalizes to

\operatorname{Tr}(\slash{A} \slash{B}) = 2^{d/2 - 1} (A \cdot B)

in some schemes, but often normalized to match four-dimensional limits). Odd-number traces still vanish for

d

even, but

\gamma_5

traces pose challenges due to its definition in non-integer dimensions; schemes like 't Hooft-Veltman treat

\gamma_5

as formally four-dimensional to preserve the epsilon structure while regularizing the remaining gammas. These adaptations ensure consistency in perturbative calculations.^[6]^[11]

Applications in Quantum Field Theory

In the Dirac Equation

The Feynman slash notation provides a compact representation for the Dirac operator in the free-particle Dirac equation, which governs the relativistic dynamics of spin-1/2 fermions. The equation is expressed as

(i \partial\!\!/ - m) \psi = 0

, where

\partial\!\!/ = \gamma^\mu \partial_\mu

contracts the gamma matrices

\gamma^\mu

with the four-gradient

\partial_\mu

, and

m

is the fermion mass.^[12] This form, introduced by Richard Feynman to streamline calculations in quantum field theory, replaces the explicit summation over Lorentz indices, enhancing readability while preserving the equation's structure.^[13] The notation underscores the equation's first-order differential nature, distinguishing it from the second-order Klein-Gordon equation for scalar fields. For plane-wave solutions, the slash notation simplifies the eigenvalue problem in momentum space. Assuming a form

\psi(x) = u(p) e^{-i p \cdot x}

for positive-energy spinors, the Dirac equation reduces to

(\slash{p} - m) u(p) = 0

, where

\slash{p} = \gamma^\mu p_\mu

and

p^\mu = (E, \mathbf{p})

satisfies the on-shell condition

p^2 = m^2

.^[12] This algebraic condition determines the spinor

u(p)

, which describes the two independent helicity states for a given momentum, facilitating the construction of normalized solutions essential for quantization.^[14] In the massless limit, the Dirac equation simplifies to the Weyl equation

i \partial\!\!/ \psi = 0

, decoupling into independent equations for left- and right-handed chiral components.^[14] Here,

\slash{p} u(p) = 0

for the corresponding Weyl spinors, reflecting the propagation of massless fermions at the speed of light with definite helicity. The slash notation highlights the equation's manifest Lorentz covariance, as the contraction

\gamma^\mu V_\mu

for any four-vector

V^\mu

transforms appropriately under Lorentz boosts and rotations, ensuring the theory's relativistic invariance without altering the underlying Dirac algebra.^[12]

With Four-Momentum

In the conventional metric signature

(+,-,-,-)

, the Feynman slash notation applied to the four-momentum

p^\mu = (E, \mathbf{p})

is defined as

\slash{p} = \gamma^\mu p_\mu = \gamma^0 E - \boldsymbol{\gamma} \cdot \mathbf{p}

, where

E

is the energy and

\mathbf{p}

is the three-momentum vector.^[15] This contraction simplifies calculations in the Dirac equation by combining the Lorentz contraction with the gamma matrices into a single operator.^[4] In the Dirac basis, where

\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}

and

\gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix}

for

i=1,2,3

with

\sigma^i

the Pauli matrices, the explicit 4×4 matrix representation of

\slash{p}

\slash{p} = \begin{pmatrix} E I_2 & -\boldsymbol{\sigma} \cdot \mathbf{p} \\ \boldsymbol{\sigma} \cdot \mathbf{p} & -E I_2 \end{pmatrix}.

This form is general and applies off-shell, with

E

independent of

\mathbf{p}

. On-shell, for massive particles where

E = \sqrt{|\mathbf{p}|^2 + m^2}

E=∣p∣2+m2

, the positive-energy spinors $u(p)$ satisfy $\slash{p}\, u(p) = m u(p)$ , serving as a projection condition for the two positive-energy solutions.^[4] The mostly-plus metric signature $(-,+,+,+)$ reverses the overall sign, yielding $\slash{p} = -\gamma^0 E + \boldsymbol{\gamma} \cdot \mathbf{p}$ and flipping the signs of the off-diagonal blocks in the matrix.^[15] For the massless case ( $m=0$ ), the on-shell condition simplifies to $E = |\mathbf{p}|$ and $\slash{p}\, u(p) = 0$ , with solutions being Weyl spinors of definite chirality. For instance, considering motion along the positive z-axis with $\mathbf{p} = (0,0,p)$ and $p > 0$ , the matrix becomes $\slash{p} = \begin{pmatrix} p & 0 & -p & 0 \\ 0 & p & 0 & p \\ p & 0 & -p & 0 \\ 0 & -p & 0 & -p \end{pmatrix},$

p0p00p0−p−p0−p00p0−p

, whose kernel yields the right-handed positive-helicity spinor $u(p) \propto \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}$

1010

(up to normalization), satisfying $\slash{p}\, u(p) = 0$ .^[15]

In Propagators and Interactions

In quantum electrodynamics (QED), the slash notation plays a central role in expressing the Dirac propagator for fermions, which describes the propagation of electrons or other charged particles between interactions. The momentum-space propagator takes the form

S(p) = \frac{i (\slash{p} + m)}{p^2 - m^2 + i\epsilon},

where

\slash{p} = \gamma^\mu p_\mu

m

is the fermion mass, and the

i\epsilon

prescription ensures the correct boundary conditions for Feynman diagrams.^[4] This compact representation leverages the algebraic properties of the slash operator to simplify the inversion of the Dirac operator in the free-field Lagrangian.^[16] At interaction vertices in QED, the slash notation facilitates the description of fermion-photon couplings, with the vertex factor given by

-i e \gamma^\mu

, where

e

is the electric charge. In some effective field theories, this extends to slashed forms such as

\slash{A}

for the photon field

A^\mu

, incorporating the contraction directly into the interaction term

\bar{\psi} (\slash{A}) \psi

.^[4] These vertex rules, combined with the slashed propagator, form the basis of Feynman diagrams for scattering amplitudes.^[16] The use of slash notation in Feynman rules significantly simplifies computations involving loop integrals, where multiple momentum slashes appear in traces and contractions. For instance, in momentum routings through loops, the notation reduces cumbersome tensor expressions to scalar-like forms amenable to dimensional regularization or other techniques.^[16] A representative example is the one-loop electron self-energy diagram in QED, depicted as a fermion line with an internal photon loop. The corresponding amplitude involves the integral

\Sigma(p) = i e^2 \mu^{4-d} \int \frac{d^d k}{(2\pi)^d} \gamma^\mu \frac{\slash{p} - \slash{k} + m}{(p - k)^2 - m^2} \gamma^\nu \frac{-i g_{\mu\nu}}{k^2 + i\epsilon},

where the slashes enable efficient evaluation via trace identities, such as

\operatorname{Tr}[\slash{a} \slash{b}] = 4 a \cdot b

, to isolate the divergent and finite contributions.^[16] This structure highlights how slash notation streamlines the Dirac algebra in perturbative expansions.^[4] The application of slash notation extends beyond QED to quantum chromodynamics (QCD), where the quark propagator mirrors the Dirac form

S(p) = i (\slash{p} + m)/(p^2 - m^2 + i\epsilon)

, but vertices incorporate color factors via

-i g_s \gamma^\mu t^a

, with

g_s

the strong coupling and

t^a

the SU(3) generators.^[17] In weak interactions, chiral projections introduce modified slashes, such as

\slash{p}_L = \slash{p} (1 - \gamma^5)/2

for left-handed currents, appearing in charged-current vertices like

\bar{u} \gamma^\mu (1 - \gamma^5) d W_\mu

.^[18] These extensions preserve the notational efficiency while accounting for color and parity-violating structures in the Standard Model.^[18]

Info Pages

Talk Pages

Special Pages

Feynman slash notation

Recent from talks

Recent from talks

Contribute something

Contribute something

Media Pages

Timelines

Articles

Notes collections

Notes

Notes

Days in Chronicle

Feynman slash notation

Identities

With four-momentum

See also

References