The question of whether brain death is defined as death is highly debated in the Halachic world, and religious circles, and carried ramifications of several Halachic issues, such as removing the patient from life support, and organ transplants. The following is an analysis on the subject and the various positions of the Poskim.
Definition and background: Brain death is the complete cessation of all brain function. It is caused due to extreme trauma or injury to the brain, which blocks the body’s blood/oxygen supply to the brain, and as a result the cells of the brain die. Once the brain is dead, its status is irreversible, and the brain can never again become function-able. Brain death, unlike a coma or vegetative state, causes many of the vital brain-dependent organs to stop, including the lungs. Nevertheless, the function of the heart is independent of the brain and so long as it receives a supply of oxygen, it will continue to function. Prior to the discovery of life support and intubation, one who suffered brain death would also have his heart stop, due to the cease of oxygen supply from his lungs. However, with the discovery and use of life support intubation, it is possible to artificially pump oxygen into the lungs, and thus allow for the continuous function of the heart. This new reality entered both the medical and Rabbinical world into a great debate as to the life/death status of a person who is brain dead, but his heart is pumping due to artificial supply of oxygen. Is the brain the definition of life, and hence one who is brain dead is considered dead irrelevant of his heart’s function, or is life determined by the heart’s function and he is hence still considered alive? Now, while it is clear that according to Halacha that the determining factor of life is one’s ability to breathe and not necessarily one’s pulse, nevertheless, it is questionable as to whether this breath must be natural or it suffices even for it to be artificial, so long as he is breathing. Practically, in the secular world it became accepted that a person who is brain dead is deemed dead for all medical and legal purposes and he may be removed from life-support by the hospital. In most states, this may be done even against the will of the family. However, in Halacha the issue still remains a great matter of debate, which touches on various serious Halachic issues when caring for a loved one, such as removal from life support, or allowing organ transplants.
The ruling: Some Poskim rule that one who is 100% brain dead is considered dead, as it is impossible for him to breathe on his own. Accordingly, one would not be allowed to desecrate Shabbos on his behalf and it would be permitted to remove him from life support [i.e. “pull the plug”]. Majority of Poskim, however, rule that he is considered alive and one may hence desecrate Shabbos on his behalf. It would likewise possibly be considered murder to remove such a person from life support, as explained in Halacha G. Another ramification between the two approaches is regarding organ transplants, which generally take place on brain dead patients, as explained in Halacha F. Practically, a Rav who is expert in the medical field is to be contacted, although in general, the approach of Chareidi Jewry is to be stringent and consider the person as alive, while the approach of modern orthodoxy is to be lenient.
Determining brain death: One of the major issues involved in determining death based on cessation of brain function is the ability to diagnose without any doubt that brain function has in truth completely ceased. Practically, there are many methods doctors use to determine brain death, primarily being the inability of the patient to breathe on his own. This can be determined through removing the respirator from the person and monitoring breathing movement. At times a patient can mildly breath on his own, but not enough to live. Such a person is not brain dead.
 See Nishmas Avraham 2:339 pp. 451-482 in great length for opinions, and correspondence letters of Gedolei Haposkim; For the most updated and thorough research on this subject, including the different approaches, supports in the Poskim, and practical rulings of the leading Rabbanim of today as interviewed by the writers-see the journal of the RCA [110 pages long] on this issue titled “Halachic Issues in the Determination of Death and in Organ Transplantation”
 Brain death versus a coma or vegetative state: Brain death is very different than a state of coma or vegetative state. A coma or vegetative state simply means the person is unconscious, asleep like, and cannot be awakened. However, the brain, as well as various of the brain dependent organs such as the lungs, still function without artificial help. Likewise, the state of the coma is reversible, and the person can one day wake up and live a healthy life. Brain death, however, is a dead brain that has no function, is irreversible, and causes brain-dependent organs to stop functioning.
 The three most vital organs on which human life is dependent is undoubtably the brain, the lungs and heart. For a person to live there needs to be a constant supply of oxygen to the cells of the body. All three of the above organs place a vital role in this supply. The function of the brain, amongst other functions, is to pump the lung. The function of the lungs is to breath the oxygen into the body. The function of the heart is to spread the oxygen, glucose, and water throughout one’s body. The brain/lung function is called respiratory while the heart function is called circulatory.
 See Admur 329:3 [follow breath]; Michaber 329:4; Yuma Mishneh 83a; Gemara 84a for a Machlokes if we follow the heart/pulse or the breath
 In the USA, this is called the Uniform Determination of Death Act (UDDA), which was passed in 1981, which gives two classes for the definition of death, one based on cessation of the respiratory function, and the second being the cessation of both the respiratory and circulatory function. This Act was based on the Harvard criteria, which was a study published by the Harvard medical school in 1968 to determine brain death as actual death.
 See http:::www.braindeath.org:law.htm for the law in your state. Most [if not all] states give medical personal the sole power of determining whether to remove the person from life support, after determining legal death due to 100% brain death. For example, the state of Texas may remove a patient from life support after determining brain death despite protests from the next of kin. However, the state of New Jersey, requires the medical personal to notify the family in case of brain death and respect their decision, whether to remove the patient from life-support or have him remain. If, however, there is a cessation of both the respiratory and circulatory functions, then he is determined dead irrelevant of the families wishes.
 Igros Moshe Y.D. 3:132; 8:54 “The doctors accepted the Halachic position of death, which is the Harvard criteria”; See Tzitz Eliezer 13:89 based on Chasam Sofer that brain-death is defined as death, in contradiction to his other responses in 10:25; For an in depth analysis as to the true opinion of the Igros Moshe-see the above journal; The Rabbanut Hareishit in 1986, under the council of Rav Mordechai Eliyahu, Rav Shapiro, Rav Zalman Nechemia Goldberg and leading physicians in the Machon Shlezinger institute of Shaareiy Tzedek, have accepted this approach. See the above journal for further discussions they had with the above Rabbanim and if any retracted their rulings.
 The reason: As we rule in Halacha that we follow the breath for determination of life [See, and not the pulse of the heart, and hence if he can no longer breathe on his own, he is considered dead even if his heart still beats. [Igros Moshe ibid] One who is brain dead is considered as if his head is removed from his body, as the brain completely rots. [Igros Moshe 8:54] See also Chasam Sofer Y.D. 338 for a Teshuvah that supports this stance.
 Tzitz Eliezer 10:25 based on Chasam Sofer and Chacham Tzvi and Rashi in Yuma ibid; Minchas Yitzchak 5:7; Ruling of Rav Elyashiv; Rav SZ”A, Rav Wozner, as detailed in Nishmas Avraham ibid and the above journal; Rav Elyashiv stated that one who can live for several months is a Safek Goses, Safek alive, and thus a brain dead patient is only viewed as a Safek Goses versus Safek alive. [Nishmas Avraham Y.D. 339 p. 469] However, Rav S”Z Aurbauch stated that a brain dead patient is viewed as a Safek Goses versus Safek dead. [Nishmas Avraham Y.D. 339 p. 467, 473, 477]
 The reason: As in actuality he is still breathing, and his heart works. It is thus not similar to the case of the Gemara in which case his breathing has stopped and cannot be returned. Alternatively, even the beating of the heart is a sign of life, as implied from Rashi ibid and Chasam Sofer. Alternatively, the reason is because it is not possible to determine 100% brain death.
 See Igros Moshe Y.D. 3:132; 8:54; See Tzitz Eliezer 13:89