Nirvikalpa is a Sanskrit adjective with the general sense of "not admitting an alternative", formed by applying the contra-existential prepositional prefix निः ni ("away, without, not") to the term विकल्प vikalpa ("alternative, variant thought or conception").
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Raja Yoga
In Raja Yoga, nirvikalpa samadhi is a synonym for Asamprajnata Samadhi, the highest stage of samadhi. Heinrich Zimmer in his book distinguishes Nirvikalpa Samadhi from other states as follows:
Nirvikalpa samādhi, on the other hand, absorption without self-consciousness, is a mergence of the mental activity (cittavṛtti) in the Self, to such a degree, or in such a way, that the distinction (vikalpa) of knower, act of knowing, and object known becomes dissolved — as waves vanish in water, and as foam vanishes into the sea.
According to Swami Sivananda, it is also called Nirbija Samadhi:
"Without seeds or Samskaras [...] All the seeds or impressions are burnt by the fire of knowledge [...] all the Samskaras and Vasanas which bring on rebirths are totally freed up. All Vrittis or mental modifications that arise from the mind-lake come under restraint. The five afflictions, viz., Avidya (ignorance), Asmita (egoism), Raga-dvesha (love and hatred) and Abhinivesha (clinging to life) are destroyed and the bonds of Karma are annihilated [...] It gives Moksha (deliverance from the wheel of births and deaths). With the advent of the knowledge of the Self, ignorance vanishes. With the disappearance of the root-cause, viz., ignorance, egoism, etc., also disappear."
Shaivism
Nirvikalpaka yoga is a technical term in the philosophical system of Shaivism, in which there is a complete identification of the "I" and Shiva, in which the very concepts of name and form disappear and Shiva alone is experienced as the real Self. In that system, this experience occurs when there is complete cessation of all thought-constructs.
Buddhism
In Buddhist philosophy, the technical term nirvikalpa-jñāna is translated by Edward Conze as "undifferentiated cognition". Conze notes that only the actual experience of nirvikalpa-jñāna can prove the reports given of it in scriptures. He describes the term as used in the Buddhist context as follows:
The "undiscriminate cognition" knows first the unreality of all objects, then realizes that without them also the knowledge itself falls to the ground, and finally directly intuits the supreme reality. Great efforts are made to maintain the paradoxical nature of this gnosis. Though without concepts, judgements and discrimination, it is nevertheless not just mere thoughtlessness. It is neither a cognition nor a non-cognition; its basis is neither thought nor non-thought.... There is here no duality of subject and object. The cognition is not different from that which is cognized, but completely identical with it.
A different sense in Buddhist usage occurs in the Sanskrit expression nirvikalpayati (Pali: nibbikappa) that means "makes free from uncertainty (or false discrimination)" = distinguishes, considers carefully.