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Formulation of quantum mechanics
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces
Path_integral_formulation
Topics referred to by the same term
Path integral may refer to: Line integral, the integral of a function along a curve Contour integral, the integral of a complex function along a curve
Path_integral
Definite integral of a scalar or vector field along a path
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear
Line_integral
Molecular dynamics simulations augmented with quantum mechanics
Path integral molecular dynamics (PIMD) is a method of incorporating quantum mechanics into molecular dynamics simulations using Feynman path integrals
Path integral molecular dynamics
Path_integral_molecular_dynamics
Operation in mathematical calculus
integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral,
Integral
Physical quantity of dimension energy × time
integrated along the path followed by the physical system. The action is typically represented as an integral over time, taken along the path of the system between
Action_(physics)
Relationship between branches of physics
This article relates the Schrödinger equation with the path integral formulation of quantum mechanics using a simple nonrelativistic one-dimensional single-particle
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
Relation_between_Schrödinger's_equation_and_the_path_integral_formulation_of_quantum_mechanics
Physics experiment
double-slit experiment can illustrate the path integral formulation of quantum mechanics provided by Feynman. The path integral formulation replaces the classical
Double-slit_experiment
Theoretical framework in physics
_{I}\rangle .} Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral: ⟨ ϕ F | e − i H T | ϕ I ⟩ = ∫ D ϕ ( t ) exp { i ∫
Quantum_field_theory
Pictorial representation of the behavior of subatomic particles
closely tied to the functional integral formulation of quantum mechanics, also invented by Feynman—see path integral formulation. The naïve application
Feynman_diagram
Calculation of strain energy release rate
J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice, who showed that an energetic contour path integral (called
J-integral
Theorem in complex analysis
U} , then the path integral ∫ γ f ( z ) d z {\displaystyle \textstyle \int _{\gamma }f(z)\,dz} is path independent for all paths in U {\displaystyle
Cauchy's_integral_theorem
Mapping involving integration between function spaces
integral transforms find special applicability within other scientific and mathematical disciplines. Another usage example is the kernel in the path integral:
Integral_transform
Partial differential equation
Rutkowski (2008). Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application
Fokker–Planck_equation
Systematic procedure of turning a classical theory into a quantum one
can also be constructed from the action of the system by means of the path integral formulation. Loop quantum gravity (loop quantization) Uncertainty principle
Quantization_(physics)
Integral of the Gaussian function, equal to sqrt(π)
of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator
Gaussian_integral
Method of analysis applied to problems wave propagation
are landing. The set of possible photon paths is consistent with Richard Feynman's path integral theory, the paths determined by the surroundings: the photon's
Huygens–Fresnel_principle
Quantum algorithm to calculate path integrals in quantum multi-body problems
Path integral Monte Carlo (PIMC) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral
Path_integral_Monte_Carlo
26-dimensional string theory
worldsheets. Bosonic string theory can be said to be defined by the path integral quantization of the Polyakov action: I 0 [ g , X ] = T 8 π ∫ M d 2 ξ
Bosonic_string_theory
Curve whose curvature changes linearly
naturally take a gradual bend shape resembling an Euler spiral. In the path integral formulation of quantum mechanics, the probability amplitude for propagation
Euler_spiral
Stochastic differential equation
{r} d\mathbf {p} ={\boldsymbol {\mu }}_{X}^{2}+3\sigma _{X}^{2}} A path integral equivalent to a Langevin equation can be obtained from the corresponding
Langevin_equation
Scientific theory
{\displaystyle M=1} . Path integral formulation comes from quantum mechanics. But for a Langevin SDE we can also induce a corresponding path integral. Considering
Langevin_dynamics
American theoretical physicist (1918–1988)
physics of elementary particles". He is also known for his work in the path integral formulation of quantum mechanics, the theory of the physics of the superfluidity
Richard_Feynman
Solitons in Euclidean spacetime
Instantons are important in quantum field theory because: they appear in the path integral as the leading quantum corrections to the classical behavior of a system
Instanton
Integration over the space of functions
probability, in the study of partial differential equations, and in the path integral formulation to the quantum mechanics of particles and fields. While
Functional_integration
Stochastic process generalizing Brownian motion
how random motion evolves over time. It also underpins the rigorous path integral formulation of quantum mechanics: by the Feynman–Kac formula, one can
Wiener_process
Description of physical properties at the atomic and subatomic scale
Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between
Quantum_mechanics
Approach to quantum gravity utilizing Wick rotations
possible at the same time. According to the Feynman path-integral formulation of quantum mechanics, the path of the quantum object is described mathematically
Euclidean_quantum_gravity
Quantum mechanical model
Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (2012) World Scientific, ISBN 978-9810-4397-5. This reference
Quantum_harmonic_oscillator
How many times curves wind around each other
simply Gauss's linking integral. This is the simplest example of a topological quantum field theory, where the path integral computes topological invariants
Linking_number
Identity in abelian theories due to gauge invariance
vacuum polarization and of the electron vertex function in QED. In the path integral formulation, the Ward–Takahashi identities are a reflection of the invariance
Ward–Takahashi_identity
Theory of quantum gravity merging quantum mechanics and general relativity
canonical theory to the path integral formulation. It turns out there are alternative routes to formulating the path integral, however their connection
Loop_quantum_gravity
Force resulting from the quantisation of a field
turn the double integral into a single integral. The q in front is the Jacobian, and the 2π comes from the angular integration. The integral converges if
Casimir_effect
Integration for Grassmann variables
properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions. Let
Berezin_integral
Relativistic quantum mechanical wave equation
to define the associated quantum field theory, such as through the path integral formulation. In quantum mechanics, the Dirac spinor ψ ( x ) {\displaystyle
Dirac_equation
Phenomenon resulting from the superposition of two waves
showed that by evaluating a path integral where all possible paths are considered, that a number of higher probability paths will emerge. In thin films
Wave_interference
Field theory of scalar fields
The Feynman diagram expansion may be obtained also from the Feynman path integral formulation. The time ordered vacuum expectation values of polynomials
Scalar_field_theory
Description of a quantum-mechanical system
include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. When these approaches
Schrödinger_equation
Generating function for quantum correlation functions
for correlation functions, making them key objects of study in the path integral formalism. They are the imaginary time versions of statistical mechanics
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Two-dimensional conformal field theory
(z_{i})}\ .} It has been difficult to define and to compute this path integral. In the path integral representation, it is not obvious that Liouville theory has
Liouville_field_theory
Product of geometric length and refractive index
path followed by light and the refractive index of the medium. For inhomogeneous optical media, the product above is generalized as a path integral as
Optical_path_length
commonly known as Feynman Integrals. In the core of Path integrals lies the concept of Functional integration. Regular integrals consist of a limiting process
Path integrals in polymer science
Path_integrals_in_polymer_science
Integral equation
theory of path tracing sometimes uses a path integral (integral over possible paths from a light source to a point) instead of the integral over possible
Rendering_equation
Type of unphysical field in quantum field theory which provides mathematical consistency
into gauge quantum field theories to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov. A
Faddeev–Popov_ghost
Function in quantum field theory showing probability amplitudes of moving particles
t;x',t')=\delta (x-x').} The propagator may also be found by using a path integral: K ( x , t ; x ′ , t ′ ) = ∫ exp [ i ℏ ∫ t ′ t L ( q ˙ , q , t ) d
Propagator
Quartic potential in quantum mechanics
mechanical context this potential served as a model for the evaluation of path integrals. or the solution of the Schrödinger equation by various methods for
Double-well_potential
Theory of quantum gauge fields on a lattice
infinite-dimensional path integral, which is computationally intractable. By working on a discrete spacetime, the path integral becomes finite-dimensional
Lattice_gauge_theory
Description of gravity using discrete values
quantum gravity Integral method Causal dynamical triangulation Causal fermion systems Causal Set Theory Covariant Feynman path integral approach Dilatonic
Quantum_gravity
Wigner distribution function in physics as opposed to in signal processing
the integral representation of ⋆ {\displaystyle \star } -products, successive operations by them have been adapted to a phase-space path integral, to
Wigner quasiprobability distribution
Wigner_quasiprobability_distribution
Mathematics of a particle physics model
creation and annihilation operators (the "canonical" formalism) by using a path integral formulation, pioneered by Feynman building on the earlier work of Dirac
Mathematical formulation of the Standard Model
Mathematical_formulation_of_the_Standard_Model
Theory of subatomic structure
both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave
String_theory
Quantum field theory of electromagnetism
Mathematically, it can be derived by a semiclassical approximation to the path integral of quantum electrodynamics. Higher-order terms can be straightforwardly
Quantum_electrodynamics
Formulation of classical mechanics
functional of the trajectory can therefore be defined as the integral of the Lagrangian along the path: S [ q ] = ∫ t 0 t 1 L ( q ( t ) , q ˙ ( t ) , t ) d t
Lagrangian_mechanics
Statistical theory
limit exists, one can talk about the field configuration space integral or path integral ⟨ f ( s ) ⟩ ( s | d ) ≡ ∫ D s f ( s ) P ( s ) . {\displaystyle
Information_field_theory
Theory of stochastic partial differential equations
(up to a topological factor) the partition function of the noise. The path integral representation of the Witten index can be achieved in three steps: (i)
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Textbook by Ramamurti Shankar
Relativistic Quantum Mechanics Path Integrals – II Derivation of the Path Integral Imaginary Time Formalism Spin and Fermion Path Integrals Summary Appendix Matrix
Principles of Quantum Mechanics
Principles_of_Quantum_Mechanics
function, integrals of loop diagrams, etc. The following Gaussian integrals are useful in calculating path integrals appearing in path integral formulation
Common integrals in quantum field theory
Common_integrals_in_quantum_field_theory
Field theory involving topological effects in physics
functions or partition functions of the system are computed by the path integral of metric-independent action functionals. For instance, in the BF model
Topological quantum field theory
Topological_quantum_field_theory
Branch of mathematics studying functions of a complex variable
(as shown in Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory
Complex_analysis
Type of field appearing in the Lagrangian
S_{\text{source}}=J\phi .} This term appears in the action in Richard Feynman's path integral formulation and is responsible for the theory interactions. In a collision
Source_field
Type of electrical device
path integral of A is equal to the enclosed B flux, just as the path integral B is equal to a constant times the enclosed current The path integral of
Toroidal inductors and transformers
Toroidal_inductors_and_transformers
Quantum mechanical phenomenon
(2012). Introduction To Quantum Mechanics: Schrodinger Equation And Path Integral (2nd ed.). Singapore: World Scientific Publishing Company. ISBN 978-981-4397-76-6
Quantum_tunnelling
Physical theory with fields invariant under the action of local "gauge" Lie groups
theory is much like that of its continuum analog: a gauge-covariant action integral that characterizes "allowable" physical situations according to the principle
Gauge_theory
Calculus on stochastic processes
disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. Stochastic integrals do NOT obey the usual chain
Stochastic_calculus
Term in general relativity
Arnowitt–Deser–Misner energy (ADM energy). The term is required to ensure the path integral (a la Hawking) for quantum gravity has the correct composition properties
Gibbons–Hawking–York boundary term
Gibbons–Hawking–York_boundary_term
Determinant in functional analysis
possible generalization, often used by physicists when using the Feynman path integral formalism in quantum field theory (QFT), uses a functional integration:
Functional_determinant
Attribute of a mathematical function
domain is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally
Residue_(complex_analysis)
Asymmetry of classical and quantum action
configurations in the path integral, one finds that contributions come in pairs with opposite signs. As a result, all path integrals vanish and a theory
Anomaly_(physics)
Interpretation of quantum mechanics
history Consistent histories Many-minds interpretation "The Garden of Forking Paths" Parallel universes in fiction The Beginning of Infinity Mathematical universe
Many-worlds_interpretation
Probabilistic algorithms to simulate quantum many-body systems
to path integral Monte Carlo, with applications similar to diffusion Monte Carlo but with some different tradeoffs. Gaussian quantum Monte Carlo Path integral
Quantum_Monte_Carlo
Topological quantum field theory
Chern–Weil theory. The action integral (path integral) of the field theory in physics is viewed as the Lagrangian integral of the Chern–Simons form and
Chern–Simons_theory
Interpretation of quantum mechanics
quantization prescription can be compared to canonical quantization and the path integral formulation, and is often referred to as Nelson's stochastic quantization
Stochastic_quantum_mechanics
Fermion path integral approach in 1+1 dimensions
light). For such a discrete motion, the Feynman path integral reduces to a sum over the possible paths. Feynman demonstrated that if each "turn" (change
Feynman_checkerboard
Vector field that is the gradient of some function
that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of
Conservative_vector_field
Wager over the solution to the black hole information paradox
the path integral over all topologically non-trivial metrics is asymptotically independent of the initial state. Thus the total path integral is unitary
Thorne–Hawking–Preskill_bet
Postulate in particle physics
Hellmann–Feynman theorem Feynman slash notation Feynman parametrization Path integral formulation Parton model Sticky bead argument One-electron universe
One-electron_universe
Fact that observing a situation changes it
Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories (path integral) Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger Interpretations
Observer_effect_(physics)
Atom of the element hydrogen
(non-relativistic) hydrogen atom was solved for the first time within Feynman's path integral formulation of quantum mechanics by Duru and Kleinert. This work greatly
Hydrogen_atom
Formulation of the principle of stationary action
Richard Feynman's path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations
Hamilton's_principle
Converting classical mechanics to quantum mechanics
gabbiano. ISBN 9788896293119. Klauder, John (2001). "Coherent State Path Integrals Without Resolutions of Unity". Foundations of Physics. 31 (1): 57–67
First_quantization
Principle suggesting that time travel paradoxes are inherently impossible
quantum mechanics, performing a quantum-mechanical sum over histories (path integral) using only the consistent extensions, and found that this resulted
Novikov self-consistency principle
Novikov_self-consistency_principle
Short "burst" or "envelope" of restricted wave action that travels as a unit
into many segments, it allows the time evolution to be expressed as a path integral. The spreading of wave packets in quantum mechanics is directly related
Wave_packet
Quantum field theory
of quantizing the Yang–Mills theory is by functional methods, i.e. path integrals. One introduces a generating functional for n-point functions as Z
Yang–Mills_theory
Approach to quantum theory
{\displaystyle S} . Although it is superficially different from the path integral formulation where the action is a classical function, the modern formulation
Schwinger's quantum action principle
Schwinger's_quantum_action_principle
Type of two-dimensional quasiparticle
classes of paths (i.e. notion of equivalence on braids) are relevant hints at a more subtle insight. It arises from the Feynman path integral, in which
Anyon
Anticommutating number
number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used
Grassmann_number
Topological structure in loop quantum gravity
configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structures are employed in loop
Spin_foam
Generalization of the concept from statistical mechanics
these theories, the partition function is heavily exploited in the path integral formulation, with great success, leading to many formulas nearly identical
Partition function (mathematics)
Partition_function_(mathematics)
Statistical principle
is sometimes called the "caliber" of the system, and is given by the path integral S [ ρ [ x ( ) ] ] = − ∫ D x ρ [ x ( ) ] ln ρ [ x ( ) ] π [ x ( ) ]
Principle_of_maximum_caliber
Quantum version of the classical action
action S [ ϕ ] {\displaystyle S[\phi ]} can be fully described in the path integral formalism using the partition functional Z [ J ] = ∫ D ϕ e i S [ ϕ ]
Effective_action
(2012). Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (2nd ed.). World Scientific. ISBN 9789814397735. Sakurai, J. J.; Napolitano
List of textbooks on classical mechanics and quantum mechanics
List_of_textbooks_on_classical_mechanics_and_quantum_mechanics
Kinematic prediction of quantum field theory for an accelerating observer
being singular. So e 2 π i H = I d . {\displaystyle e^{2\pi iH}=Id.} A path integral with real time coordinate is dual to a thermal partition function, related
Unruh_effect
Wave equations respecting special and general relativity
operator Ĥ describing the quantum system. Alternatively, Feynman's path integral formulation uses a Lagrangian rather than a Hamiltonian operator. More
Relativistic_wave_equations
heat bath. In particular, in the path integral representation of a Euclidean quantum field theory, the path integral measure is closely related to the
Stochastic_quantization
Formulation to quantize gauge field theories in physics
states when performing QFT calculations. Crucially, this symmetry of the path integral is preserved in loop order, and thus prevents introduction of counterterms
BRST_quantization
Principle of quantum mechanics
amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that
Quantum_superposition
Quantum mechanics taking into account particles near or at the speed of light
in a non-relativistic background, a few of them (e.g. the Dirac or path-integral formalism) also work with special relativity. Key features common to
Relativistic quantum mechanics
Relativistic_quantum_mechanics
Framework to describe phase transitions
mechanics of fields, and shares with it many techniques, such as the path integral formulation and renormalization. If the system involves polymers, it
Statistical_field_theory
Quantum field theory with four-point interactions
unbounded below, and there would be no stable vacuum. Also, the Feynman path integral discussed below would be ill-defined. In 4 dimensions, ϕ 4 {\displaystyle
Quartic_interaction
Theory of gravitation as curved spacetime
on the Feynman Path Integral approach and Regge calculus, dynamical triangulations, causal sets, twistor models or the path integral based models of
General_relativity
PATH INTEGRAL
PATH INTEGRAL
Surname or Lastname
English (mainly Devon)
English (mainly Devon) : variant of Pate 1.
Surname or Lastname
Scottish
Scottish : reduced form of McGath.English : variant of Garth.North German (Gäth) : variant of Gäde (see Gaede).North German : topographic name from Middle Low German gate ‘street’, ‘alley’.
Surname or Lastname
English
English : habitational name from the city of Bath in western England, which is the site of sumptuous, but in the Middle Ages ruined, Roman baths. The place is named with the dative plural of Old English bæð ‘bath’. In some cases the surname may have originated as a metonymic occupational name for an attendant at a public bath house.Scottish : reduced and altered form of McBeth.German : variant of Bathe.Indian (Panjab) : Sikh name based on the name of a Jat clan.
Female
Hebrew
(בַּת-ש×ֶבַע) Hebrew name BATH-SHEBA means "daughter of the oath." In the bible, this is the name of a wife of Uriah then later King David, and mother of Solomon. Also spelled Bat-Sheva, Bathsheba, and Bathsheva.
Male
Irish
Short form of Irish Gaelic Parthalán, possibly PARTH means "son of Talmai."
Female
English
Short form of English Katherine, KATH means "pure."
Female
Hebrew
(בַּתש×וּעַ) Variant spelling of Hebrew Bath-Shuwa, BATH-SHUA means "daughter of wealth."Â
Boy/Male
Muslim/Islamic
Correct path Straight path
Boy/Male
Indian
Victory
Surname or Lastname
English (Bath)
English (Bath) : unexplained.
Female
Hebrew
(×Ö¸×¡Ö°× Ö·×ª) Hebrew name of Egyptian origin, ACÄ”NATH means "belonging to the goddess Neith." In the bible, this is the name of Joseph's Egyptian wife.
Boy/Male
Arabic, Australian, Muslim
Correct Path; Straight Path
Female
Hebrew
(בַּתש×וּעַ) Hebrew name BATH-SHUWA means "daughter of wealth." In the bible, this is another name Bath-Sheba is known by.
Female
English
English short form of French Catherine, CATH means "pure."
Male
English
English unisex short form of English Patrick and Latin Patricia, PAT means "patrician; of noble birth."
Girl/Female
Australian, British, English
Way
Boy/Male
Muslim
Correct path, Straight path
Boy/Male
Arabic, Modern
Road; The Way
Surname or Lastname
English and Scottish
English and Scottish : from the personal name Pat(t), Pate, a short form of Patrick.English and Scottish : nickname for a man with a bald head, from Middle English pate ‘head’, ‘skull’.French (Paté) : from Old French pat(t)é ‘with paws’, ‘pawed’ (from pat(t)e ‘paw’), a nickname, applied presumably to a man with large and clumsy hands and feet.German : nickname for a trustworthy man, from Middle High German pate, Middle Low German pade ‘godfather’, ‘male relative’ (see Paeth), or alternatively from a personal name Bado, probably meaning ‘battle’, ‘fight’.
Surname or Lastname
English (Bristol and Bath)
English (Bristol and Bath) : unexplained.
PATH INTEGRAL
PATH INTEGRAL
Girl/Female
German
Noble; Kind
Boy/Male
Tamil
Calm
Boy/Male
Hindu, Indian, Tamil
Lord Murugan
Female
English
Variant spelling of English Laverne, possibly LAVERN means "spring-like; to be verdant." Compare with masculine Lavern.
Female
Native American
Native American Sioux name HANTAYWEE means "faithful."
Boy/Male
Shakespearean
Pericles, Prince of Tyre' Lord of Tyre.
Boy/Male
Arabic
Confident; Strong
Boy/Male
Hindu, Indian
Water
Boy/Male
American, British, English, Latin
To Endure; Contemporary Phonetic Variant of Dante; Enduring
Girl/Female
Tamil
Life, Immortal
PATH INTEGRAL
PATH INTEGRAL
PATH INTEGRAL
PATH INTEGRAL
PATH INTEGRAL
pl.
of Path
n.
A way, course, or track, in which anything moves or has moved; route; passage; an established way; as, the path of a meteor, of a caravan, of a storm, of a pestilence. Also used figuratively, of a course of life or action.
v. t.
To adorn, as the face, with a patch or patches.
v. t.
To mend by sewing on a piece or pieces of cloth, leather, or the like; as, to patch a coat.
n.
Way; track; path.
v. t.
To make a path in, or on (something), or for (some one).
pr.p. & vb. n.
of Path
v. t.
To mend with pieces; to repair with pieces festened on; to repair clumsily; as, to patch the roof of a house.
n.
The act of exposing the body, or part of the body, for purposes of cleanliness, comfort, health, etc., to water, vapor, hot air, or the like; as, a cold or a hot bath; a medicated bath; a steam bath; a hip bath.
n.
Fig.: Anything regarded as a patch; a small piece of ground; a tract; a plot; as, scattered patches of trees or growing corn.
imp. & p. p.
of Path
v. t.
To make of pieces or patches; to repair as with patches; to arrange in a hasty or clumsy manner; -- generally with up; as, to patch up a truce.
adv.
In a pat manner.
n.
A small piece of anything used to repair a breach; as, a patch on a kettle, a roof, etc.
n.
Way; road; path.
n.
Hence: The which contains the strength of life; the vital or essential part; concentrated force; vigor; strength; importance; as, the speech lacked pith.
n.
A small mass, as of butter, shaped by pats.
n.
A towing path.