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<title>Path-ordering</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Path-ordering</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">This article is about rearranging a product of operators in physics. For the well-orderings on mathematical terms, see <a href="Path_ordering_(term_rewriting)" title="Path ordering (term rewriting)">Path ordering (term rewriting)</a>.</div>
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<p>In <a href="Theoretical_physics" title="Theoretical physics">theoretical physics</a>, <b>path-ordering</b> is the procedure (or a <a href="Meta-operator" title="Meta-operator">meta-operator</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>
</semantics>
</math></span><img src="./10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"></span>) that orders a product of operators according to the value of a chosen <a href="Parameter" title="Parameter">parameter</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left\{O_{1}(\sigma _{1})O_{2}(\sigma _{2})\cdots O_{N}(\sigma _{N})\right\}\equiv O_{p_{1}}(\sigma _{p_{1}})O_{p_{2}}(\sigma _{p_{2}})\cdots O_{p_{N}}(\sigma _{p_{N}}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
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</mrow>
<mrow>
<mo>{</mo>
<mrow>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo stretchy="false">)</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
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</msub>
</mrow>
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<mi>σ<!-- σ --></mi>
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<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
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<mo stretchy="false">)</mo>
<mo>⋯<!-- ⋯ --></mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
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<mo stretchy="false">(</mo>
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<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mi>p</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left\{O_{1}(\sigma _{1})O_{2}(\sigma _{2})\cdots O_{N}(\sigma _{N})\right\}\equiv O_{p_{1}}(\sigma _{p_{1}})O_{p_{2}}(\sigma _{p_{2}})\cdots O_{p_{N}}(\sigma _{p_{N}}).}</annotation>
</semantics>
</math></span><img src="./79a800ec76e580884c480b9c59088ac8bb4b8c9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.594ex; height:3.009ex;" alt="{\displaystyle {\mathcal {P}}\left\{O_{1}(\sigma _{1})O_{2}(\sigma _{2})\cdots O_{N}(\sigma _{N})\right\}\equiv O_{p_{1}}(\sigma _{p_{1}})O_{p_{2}}(\sigma _{p_{2}})\cdots O_{p_{N}}(\sigma _{p_{N}}).}" loading="lazy"></span></dd></dl>
<p>Here <i>p</i> is a <a href="Permutation" title="Permutation">permutation</a> that orders the parameters by value:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:\{1,2,\dots ,N\}\to \{1,2,\dots ,N\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>:</mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>N</mi>
<mo fence="false" stretchy="false">}</mo>
<mo stretchy="false">→<!-- → --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>N</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p:\{1,2,\dots ,N\}\to \{1,2,\dots ,N\}}</annotation>
</semantics>
</math></span><img src="./a8a0184a268a4ba862cda1bf0c6a8207a330cf0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:32.661ex; height:2.843ex;" alt="{\displaystyle p:\{1,2,\dots ,N\}\to \{1,2,\dots ,N\}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{p_{1}}\leq \sigma _{p_{2}}\leq \cdots \leq \sigma _{p_{N}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
</msub>
<mo>≤<!-- ≤ --></mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>≤<!-- ≤ --></mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msub>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{p_{1}}\leq \sigma _{p_{2}}\leq \cdots \leq \sigma _{p_{N}}.}</annotation>
</semantics>
</math></span><img src="./5c565f1f8df03bba63e4165a3da3597d85c428f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.837ex; height:2.676ex;" alt="{\displaystyle \sigma _{p_{1}}\leq \sigma _{p_{2}}\leq \cdots \leq \sigma _{p_{N}}.}" loading="lazy"></span></dd></dl>
<p>For example:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left\{O_{1}(4)O_{2}(2)O_{3}(3)O_{4}(1)\right\}=O_{4}(1)O_{2}(2)O_{3}(3)O_{1}(4).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">P</mi>
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<mo>{</mo>
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<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>}</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<msub>
<mi>O</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left\{O_{1}(4)O_{2}(2)O_{3}(3)O_{4}(1)\right\}=O_{4}(1)O_{2}(2)O_{3}(3)O_{1}(4).}</annotation>
</semantics>
</math></span><img src="./1cc9c8c5153052425c19312ae46258ed5b10fc99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.555ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left\{O_{1}(4)O_{2}(2)O_{3}(3)O_{4}(1)\right\}=O_{4}(1)O_{2}(2)O_{3}(3)O_{1}(4).}" loading="lazy"></span></dd></dl>
<p>In many fields of physics, the most common type of path-ordering is <b>time-ordering</b>, which is discussed in detail below.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<p>If an <a href="Operator_(physics)" title="Operator (physics)">operator</a> is not simply expressed as a product, but as a function of another operator, we must first perform a <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> of this function. This is the case of the <a href="Wilson_loop" title="Wilson loop">Wilson loop</a>, which is defined as a <a href="Path-ordered_exponential" class="mw-redirect" title="Path-ordered exponential">path-ordered exponential</a> to guarantee that the Wilson loop encodes the <a href="Holonomy" title="Holonomy">holonomy</a> of the <a href="Gauge_connection" class="mw-redirect" title="Gauge connection">gauge connection</a>. The parameter <i>σ</i> that determines the ordering is a parameter describing the <a href="Contour_integration" title="Contour integration">contour</a>, and because the contour is closed, the Wilson loop must be defined as a <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> in order to be <a href="Gauge-invariant" class="mw-redirect" title="Gauge-invariant">gauge-invariant</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Time_ordering">Time ordering</h2></div>
<p>In <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> it is useful to take the <b>time-ordered</b> product of operators. This operation is denoted by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation>
</semantics>
</math></span><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy"></span>. (Although <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation>
</semantics>
</math></span><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy"></span> is often called the "time-ordering operator", strictly speaking it is neither an <a href="Linear_operator" class="mw-redirect" title="Linear operator">operator</a> on states nor a <a href="Superoperator" title="Superoperator">superoperator</a> on operators.)
</p><p>For two operators <i>A</i>(<i>x</i>) and <i>B</i>(<i>y</i>) that depend on spacetime locations x and y we define:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}\left\{A(x)B(y)\right\}:={\begin{cases}A(x)B(y)&amp;{\text{if }}\tau _{x}>\tau _{y},\\\pm B(y)A(x)&amp;{\text{if }}\tau _{x}<\tau _{y}.\end{cases}}}">
<semantics>
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<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
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<mo>}</mo>
</mrow>
<mo>:=</mo>
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<mtr>
<mtd>
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if&nbsp;</mtext>
</mrow>
<msub>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>&gt;</mo>
<msub>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>±<!-- ± --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>if&nbsp;</mtext>
</mrow>
<msub>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>&lt;</mo>
<msub>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}\left\{A(x)B(y)\right\}:={\begin{cases}A(x)B(y)&amp;{\text{if }}\tau _{x}&gt;\tau _{y},\\\pm B(y)A(x)&amp;{\text{if }}\tau _{x}&lt;\tau _{y}.\end{cases}}}</annotation>
</semantics>
</math></span><img src="./f616be5d3408c7f203ea6fd9500d755ba30c2591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.179ex; height:6.176ex;" alt="{\displaystyle {\mathcal {T}}\left\{A(x)B(y)\right\}:={\begin{cases}A(x)B(y)&amp;{\text{if }}\tau _{x}>\tau _{y},\\\pm B(y)A(x)&amp;{\text{if }}\tau _{x}<\tau _{y}.\end{cases}}}" loading="lazy"></span></dd></dl>
<p>Here <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau _{x}}</annotation>
</semantics>
</math></span><img src="./37a5ac03db3195b6344143fd455813a20b98aa6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.189ex; height:2.009ex;" alt="{\displaystyle \tau _{x}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>τ<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau _{y}}</annotation>
</semantics>
</math></span><img src="./4dd30ece77dcd554d540234729f6212aeabeb6ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.065ex; height:2.343ex;" alt="{\displaystyle \tau _{y}}" loading="lazy"></span> denote the <i>invariant</i> scalar time-coordinates of the points x and y.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Explicitly we have
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}\left\{A(x)B(y)\right\}:=\theta (\tau _{x}-\tau _{y})A(x)B(y)\pm \theta (\tau _{y}-\tau _{x})B(y)A(x),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
</mrow>
</mrow>
<mrow>
<mo>{</mo>
<mrow>
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
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<mo stretchy="false">)</mo>
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<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}\left\{A(x)B(y)\right\}:=\theta (\tau _{x}-\tau _{y})A(x)B(y)\pm \theta (\tau _{y}-\tau _{x})B(y)A(x),}</annotation>
</semantics>
</math></span><img src="./d30a9cfaaa2cbc80cd28ca590783a01727cdf2dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:60.701ex; height:3.009ex;" alt="{\displaystyle {\mathcal {T}}\left\{A(x)B(y)\right\}:=\theta (\tau _{x}-\tau _{y})A(x)B(y)\pm \theta (\tau _{y}-\tau _{x})B(y)A(x),}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>θ<!-- θ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>
</semantics>
</math></span><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"></span> denotes the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> and the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>±<!-- ± --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \pm }</annotation>
</semantics>
</math></span><img src="./869e366caf596564de4de06cb0ba124056d4064b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \pm }" loading="lazy"></span> depends on if the operators are <a href="Boson" title="Boson">bosonic</a> or <a href="Fermion" title="Fermion">fermionic</a> in nature. If bosonic, then the + sign is always chosen, if fermionic then the sign will depend on the number of operator interchanges necessary to achieve the proper time ordering. Note that the statistical factors do not enter here.
</p><p>Since the operators depend on their location in spacetime (i.e. not just time) this time-ordering operation is only coordinate independent if operators at <a href="Spacelike" class="mw-redirect" title="Spacelike">spacelike</a> separated points <a href="Commutativity" class="mw-redirect" title="Commutativity">commute</a>. This is why it is necessary to use <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>τ<!-- τ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>
</semantics>
</math></span><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"></span> rather than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>
</semantics>
</math></span><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"></span>, since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>
</semantics>
</math></span><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"></span> usually indicates the coordinate dependent time-like index of the spacetime point. Note that the time-ordering is usually written with the time argument increasing from right to left.
</p><p>In general, for the product of <i>n</i> field operators <span class="nowrap"><i>A</i><sub>1</sub>(<i>t</i><sub>1</sub>), …, <i>A</i><sub><i>n</i></sub>(<i>t</i><sub><i>n</i></sub>)</span> the time-ordered product of operators are defined as follows:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {T}}\{A_{1}(t_{1})A_{2}(t_{2})\cdots A_{n}(t_{n})\}&amp;=\sum _{p}\theta (t_{p_{1}}>t_{p_{2}}>\cdots >t_{p_{n}})\varepsilon (p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t_{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})\\&amp;=\sum _{p}\left(\prod _{j=1}^{n-1}\theta (t_{p_{j}}-t_{p_{j+1}})\right)\varepsilon (p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t_{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {T}}\{A_{1}(t_{1})A_{2}(t_{2})\cdots A_{n}(t_{n})\}&amp;=\sum _{p}\theta (t_{p_{1}}&gt;t_{p_{2}}&gt;\cdots &gt;t_{p_{n}})\varepsilon (p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t_{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})\\&amp;=\sum _{p}\left(\prod _{j=1}^{n-1}\theta (t_{p_{j}}-t_{p_{j+1}})\right)\varepsilon (p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t_{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./b5730877ca620a546cb71e21ebf80593d96d152a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.952ex; margin-bottom: -0.22ex; width:90.77ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {T}}\{A_{1}(t_{1})A_{2}(t_{2})\cdots A_{n}(t_{n})\}&amp;=\sum _{p}\theta (t_{p_{1}}>t_{p_{2}}>\cdots >t_{p_{n}})\varepsilon (p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t_{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})\\&amp;=\sum _{p}\left(\prod _{j=1}^{n-1}\theta (t_{p_{j}}-t_{p_{j+1}})\right)\varepsilon (p)A_{p_{1}}(t_{p_{1}})A_{p_{2}}(t_{p_{2}})\cdots A_{p_{n}}(t_{p_{n}})\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>where the sum runs all over <i>p'</i>s and over the <a href="Symmetric_group" title="Symmetric group">symmetric group</a> of <i>n</i> degree permutations and
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (p)\equiv {\begin{cases}1&amp;{\text{for bosonic operators,}}\\{\text{sign of the permutation}}&amp;{\text{for fermionic operators.}}\end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ε<!-- ε --></mi>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>≡<!-- ≡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>{</mo>
<mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>for bosonic operators,</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>sign of the permutation</mtext>
</mrow>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>for fermionic operators.</mtext>
</mrow>
</mtd>
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<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varepsilon (p)\equiv {\begin{cases}1&amp;{\text{for bosonic operators,}}\\{\text{sign of the permutation}}&amp;{\text{for fermionic operators.}}\end{cases}}}</annotation>
</semantics>
</math></span><img src="./117110040c3bad116cc7179ceeecc5313d1f4c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.804ex; height:6.176ex;" alt="{\displaystyle \varepsilon (p)\equiv {\begin{cases}1&amp;{\text{for bosonic operators,}}\\{\text{sign of the permutation}}&amp;{\text{for fermionic operators.}}\end{cases}}}" loading="lazy"></span></dd></dl>
<p>The <a href="S-matrix" title="S-matrix">S-matrix</a> in <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> is an example of a time-ordered product. The S-matrix, transforming the state at <span class="nowrap"><i>t</i> = −∞</span> to a state at <span class="nowrap"><i>t</i> = +∞</span>, can also be thought of as a kind of "<a href="Holonomy" title="Holonomy">holonomy</a>", analogous to the <a href="Wilson_loop" title="Wilson loop">Wilson loop</a>. We obtain a time-ordered expression because of the following reason:
</p><p>We start with this simple formula for the exponential
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp h=\lim _{N\to \infty }\left(1+{\frac {h}{N}}\right)^{N}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>h</mi>
<mo>=</mo>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</munder>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>h</mi>
<mi>N</mi>
</mfrac>
</mrow>
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<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \exp h=\lim _{N\to \infty }\left(1+{\frac {h}{N}}\right)^{N}.}</annotation>
</semantics>
</math></span><img src="./e39c8d9ce029d9bf8824b32155e06dcb99526e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.172ex; height:6.676ex;" alt="{\displaystyle \exp h=\lim _{N\to \infty }\left(1+{\frac {h}{N}}\right)^{N}.}" loading="lazy"></span></dd></dl>
<p>Now consider the discretized <a href="Evolution_operator" class="mw-redirect" title="Evolution operator">evolution operator</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\cdots (1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_{0})(1+h_{-1})(1+h_{-2})\cdots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>⋯<!-- ⋯ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S=\cdots (1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_{0})(1+h_{-1})(1+h_{-2})\cdots }</annotation>
</semantics>
</math></span><img src="./13da67c6e97d7a4e5081ec1c3c6cb020ba749a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.443ex; height:2.843ex;" alt="{\displaystyle S=\cdots (1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_{0})(1+h_{-1})(1+h_{-2})\cdots }" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+h_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1+h_{j}}</annotation>
</semantics>
</math></span><img src="./1fc82e5faa3b967dd57e8f7ea2911e9b7a2ab04d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.252ex; height:2.843ex;" alt="{\displaystyle 1+h_{j}}" loading="lazy"></span> is the evolution operator over an infinitesimal time interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [j\varepsilon ,(j+1)\varepsilon ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>j</mi>
<mi>ε<!-- ε --></mi>
<mo>,</mo>
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>ε<!-- ε --></mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [j\varepsilon ,(j+1)\varepsilon ]}</annotation>
</semantics>
</math></span><img src="./0e1e8b90a6749fbded80a5ce72a550cb9c43f5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.223ex; height:2.843ex;" alt="{\displaystyle [j\varepsilon ,(j+1)\varepsilon ]}" loading="lazy"></span>. The higher order terms can be neglected in the limit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \to 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ε<!-- ε --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varepsilon \to 0}</annotation>
</semantics>
</math></span><img src="./f0a6823c23666f99317e232cf7d02df6d9c9b7a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.86ex; height:2.176ex;" alt="{\displaystyle \varepsilon \to 0}" loading="lazy"></span>. The operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{j}}</annotation>
</semantics>
</math></span><img src="./614689f15f73ad5b4a5d7fa837a72614202b0d89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.249ex; height:2.843ex;" alt="{\displaystyle h_{j}}" loading="lazy"></span> is defined by
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{j}={\frac {1}{i\hbar }}\int _{j\varepsilon }^{(j+1)\varepsilon }\,dt\int d^{3}x\,H({\vec {x}},t).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>i</mi>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>ε<!-- ε --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>ε<!-- ε --></mi>
</mrow>
</msubsup>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
<mo>∫<!-- ∫ --></mo>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
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<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{j}={\frac {1}{i\hbar }}\int _{j\varepsilon }^{(j+1)\varepsilon }\,dt\int d^{3}x\,H({\vec {x}},t).}</annotation>
</semantics>
</math></span><img src="./8f8bd7f978d7819f8c3c0f27bde0a513533c9159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.709ex; height:6.676ex;" alt="{\displaystyle h_{j}={\frac {1}{i\hbar }}\int _{j\varepsilon }^{(j+1)\varepsilon }\,dt\int d^{3}x\,H({\vec {x}},t).}" loading="lazy"></span></dd></dl>
<p>Note that the evolution operators over the "past" time intervals appears on the right side of the product. We see that the formula is analogous to the identity above satisfied by the exponential, and we may write
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\mathcal {T}}\exp \left(\sum _{j=-\infty }^{\infty }h_{j}\right)={\mathcal {T}}\exp \left(\int dt\,d^{3}x\,{\frac {H({\vec {x}},t)}{i\hbar }}\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
</mrow>
</mrow>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
</mrow>
</mrow>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>∫<!-- ∫ --></mo>
<mi>d</mi>
<mi>t</mi>
<mspace width="thinmathspace"></mspace>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>i</mi>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S={\mathcal {T}}\exp \left(\sum _{j=-\infty }^{\infty }h_{j}\right)={\mathcal {T}}\exp \left(\int dt\,d^{3}x\,{\frac {H({\vec {x}},t)}{i\hbar }}\right).}</annotation>
</semantics>
</math></span><img src="./a4b40719a8b6d9fb432fb9924a31f71930aa911d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.634ex; height:7.676ex;" alt="{\displaystyle S={\mathcal {T}}\exp \left(\sum _{j=-\infty }^{\infty }h_{j}\right)={\mathcal {T}}\exp \left(\int dt\,d^{3}x\,{\frac {H({\vec {x}},t)}{i\hbar }}\right).}" loading="lazy"></span></dd></dl>
<p>The only subtlety we had to include was the time-ordering operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation>
</semantics>
</math></span><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy"></span> because the factors in the product defining <i>S</i> above were time-ordered, too (and operators do not commute in general) and the operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">T</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation>
</semantics>
</math></span><img src="./8236d074e42310f5dc24d1d2b5b8f5981c3e87ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}" loading="lazy"></span> ensures that this ordering will be preserved.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Ordered_exponential" title="Ordered exponential">Ordered exponential</a> (essentially the same concept)</li>
<li><a href="Dyson_series" title="Dyson series">Dyson series</a></li>
<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>
<li><a href="S-matrix" title="S-matrix">S-matrix</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a>, <i>The Quantum Theory of Fields</i>, Vol. 3, Cambridge University Press, 1995, <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>0-521-55001-7</bdi>, p. 143.</span>
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This article is issued from <a class="external text" title="Last edited on 2024-09-07" href="https://en.wikipedia.org/wiki/?title=Path-ordering&amp;oldid=1244434576">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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